#6848289

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=hi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6755485);