#256555

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

741 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)(3,8)(4,6)(5,7)(9,10,13,17)(11,12,20,15)(14,19,23,18)(16,24,21,22)(25,26,28,32)(27,34,35,30)(29,31,37,33)(36,40,38,39);(1,3,5,4)(2,6,7,8)(9,11,13,20)(10,15,17,12)(14,16,23,21)(18,24,19,22)(25,27,28,35)(26,30,32,34)(29,38,37,36)(31,40,33,39);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,18)(13,15)(14,24)(17,21)(19,20)(22,23)(26,31)(29,37)(30,39)(32,33)(34,40)(36,38);(2,7)(6,8)(10,17)(12,15)(18,19)(22,24)(26,32)(30,34)(31,33)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163751);