#479963

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	#97
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

551 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,6,4,5,7,8)(9,10)(11,19,20,31)(12,23)(13,18)(14,16)(15,22,27,34)(17,25)(21,35,33,32)(24,38)(26,29)(28,40,37,36)(30,39);(9,11,12,21,13,22,24,36,14,20,25,33,26,34,39,40)(10,15,17,28,18,19,30,32,16,27,23,37,29,31,38,35);(2,5)(6,8);(10,16)(11,20)(17,23)(18,29)(21,33)(22,34)(30,38)(36,40)

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,b^a=bh,c^a=cg,d^a=di,f^a=fi,d^b=di,e^b=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286688, #72286716, #72286731, …
maximal normal subgroup of	#72286688, #72286716, #72286731, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387159);