#513076

order	512
nilpotency class	5
isoclinism	isoclinic class of #512941
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	#13697	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

327 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 118 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)(11,21)(12,26)(13,27)(14,30)(16,32)(18,40)(19,41)(20,39)(22,29)(23,49)(24,52)(25,46)(28,48)(31,55)(33,37)(34,56)(35,64)(36,65)(38,66)(42,61)(43,44)(45,50)(47,69)(51,70)(53,54)(57,58)(59,63)(60,71)(62,72)(67,68);(2,5)(6,8)(9,10)(11,16)(12,18)(13,19)(14,20)(15,17)(21,32)(22,33)(23,35)(24,36)(25,34)(26,40)(27,41)(28,42)(29,37)(30,39)(31,38)(43,57)(44,58)(45,59)(46,56)(47,60)(48,61)(49,64)(50,63)(51,62)(52,65)(53,67)(54,68)(55,66)(69,71)(70,72);(9,11,22,47,13,23,28,45,30,50,54,49,55,69,12,21,15,25,26,44,31,51,53,52,14,24,48,70,27,43,29,46)(10,16,33,60,19,35,42,59,39,63,68,64,66,71,18,32,17,34,40,58,38,62,67,65,20,36,61,72,41,57,37,56);(1,3,4,7)(2,6,5,8)(10,17)(16,34)(18,37)(19,38)(20,39)(32,56)(33,40)(35,62)(36,63)(41,66)(42,67)(57,71)(58,60)(59,65)(61,68)(64,72)

Polycyclic presentation:

2^9|c^2=fgi,d^2=e,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232017, #72321790, #72322380, …
maximal normal subgroup of	#72232017, #72321790, #72322380, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420272);