#6865026

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1250 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,d^2=h,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6772222);