#297199

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	#46017	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

753 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)(11,18)(12,22)(13,16)(14,17)(15,21)(19,20)(23,24)(25,26)(27,33)(28,30)(29,37)(31,35)(32,36)(34,39)(38,40);(2,5)(6,7)(9,10)(11,19)(12,16)(13,22)(14,17)(15,21)(18,20)(23,24)(26,30)(27,34)(29,36)(31,38);(9,11,12,21,14,20,13,23)(10,15,16,18,17,24,22,19)(25,27,29,35,28,34,36,40)(26,31,32,33,30,38,37,39);(1,3,4,8)(2,6,5,7)(11,20)(15,24)(18,19)(21,23)(26,30)(31,38)(32,37)(33,39)

Polycyclic presentation:

2^9|a^2=ei,c^2=f,d^2=ei,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, 204395);