#6848321

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1349 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)(27,31)(28,34)(29,33)(30,35)(32,36)(37,40)(38,39);(3,4)(6,7)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,30,37)(26,31,35,40)(28,38,29,36)(32,34,39,33);(1,3,5,4)(2,6,8,7)(25,27)(26,32)(28,36)(29,38)(30,37)(31,33)(34,40)(35,39);(10,15)(14,17)(16,23)(22,24)(25,27)(26,31)(28,38)(29,36)(30,37)(32,33)(34,39)(35,40);(1,4,5,3)(2,7,8,6)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28,30,29)(26,33,35,34)(27,36,37,38)(31,39,40,32)

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,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6755517);