#512987

order	512
nilpotency class	5
isoclinism	isoclinic class of #512832
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33610	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

369 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,16,22)(11,17,26,44)(12,19,15,39)(13,30,35,63)(14,36,38,68)(18,45,49,24)(20,51,52,33)(21,29,53,32)(23,40,27,43)(25,55,56,69)(28,59,34,66)(31,41,64,71)(37,42,58,72)(46,57,48,54)(47,67,70,60)(50,65,62,61);(2,5)(6,8)(9,11)(10,17)(12,27)(13,24)(14,25)(15,23)(16,26)(18,30)(19,43)(20,41)(21,42)(22,44)(28,57)(29,58)(31,33)(32,37)(34,54)(35,45)(36,55)(38,56)(39,40)(46,66)(47,62)(48,59)(49,63)(50,70)(51,64)(52,71)(53,72)(60,65)(61,67)(68,69);(1,3,4,7)(2,6,5,8)(9,11)(10,18)(12,23)(13,31)(14,37)(15,27)(16,26)(17,30)(19,46)(20,42)(21,41)(22,49)(24,33)(25,32)(28,60)(29,56)(34,67)(35,64)(36,47)(38,58)(39,48)(40,59)(43,66)(44,63)(45,51)(50,69)(52,72)(53,71)(54,61)(55,62)(57,65)(68,70);(9,12,16,15)(10,19,22,39)(11,23,26,27)(13,28,35,34)(14,29,38,32)(17,40,44,43)(18,46,49,48)(20,50,52,62)(21,36,53,68)(24,54,45,57)(25,37,56,58)(30,59,63,66)(31,60,64,67)(33,61,51,65)(41,47,71,70)(42,69,72,55)

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fg,g^a=gh,h^a=hi,d^b=di,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260454, #72322387, #72322440
maximal normal subgroup of	#72260454, #72322387, #72322440

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420183);