#7185167

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1218 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=ghi,c^2=i,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092363);