#412701

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1092 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=e,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=di,g^c=gi

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, 319897);