#6847939

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1395 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)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,32,29)(27,30,34,31)(28,35,39,36)(33,40,37,38);(3,4)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,32,34)(26,30,29,31)(28,36,39,35)(33,40,37,38);(1,3,5,4)(2,6,8,7)(25,28)(26,33)(27,35)(29,37)(30,38)(31,40)(32,39)(34,36);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,32,26)(27,30,34,31)(28,37,39,33)(35,38,36,40);(1,4,5,3)(2,7,8,6)(11,19)(14,17)(18,24)(22,23)(25,30,32,31)(26,34,29,27)(28,38,39,40)(33,36,37,35)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=hi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6755135);