#297176

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

666 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:

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

Polycyclic presentation:

2^9|a^2=ehi,c^2=f,d^2=e,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,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204372);