#500068

order	512
nilpotency class	4
isoclinism	isoclinic class of #499795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#32899	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=ei,d^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,h^c=hi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317536, #72317591, #72317605, …
maximal normal subgroup of	#72317536, #72317591, #72317605, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407264);