#500529

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×H(5,𝔽₂)	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

919 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)(3,7)(4,5)(6,8)(9,10,14,18)(11,15,20,28)(12,22,24,37)(13,17,26,33)(16,30,31,23)(19,34,35,40)(21,29,25,32)(27,38,39,36);(2,5)(6,7)(9,11,14,20)(10,15,18,28)(12,19,24,35)(13,25,26,21)(16,27,31,39)(17,32,33,29)(22,34,37,40)(23,36,30,38);(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,22)(18,31)(20,35)(21,36)(25,38)(26,30)(28,39)(29,34)(32,40)(33,37);(1,3,4,8)(2,6,5,7)(11,20)(15,28)(19,35)(21,25)(27,39)(29,32)(34,40)(36,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=e,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407725);