#500231

order	512
nilpotency class	4
isoclinism	isoclinic class of #499803
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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	C₂×#535	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1107 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,19,22,33)(12,16,23,29)(13,17,25,31)(15,26,28,21)(20,32,34,40)(24,30,36,39)(27,37,38,35);(2,5)(6,8)(12,23)(13,25)(16,29)(17,31)(20,34)(27,38)(32,40)(35,37);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,27)(17,19)(18,28)(23,34)(24,35)(25,26)(29,38)(30,32)(31,33)(36,37)(39,40);(2,5)(6,8)(9,12,14,23)(10,16,18,29)(11,20,22,34)(13,24,25,36)(15,27,28,38)(17,30,31,39)(19,32,33,40)(21,35,26,37)

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,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	#72317532, #72317587, #72317620, …
maximal normal subgroup of	#72317532, #72317587, #72317620, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407427);