#7036729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1217 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6943925);