#513152

order	512
nilpotency class	5
isoclinism	isoclinic class of #513092
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	#2215	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 112 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420348);