#7206234

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,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, 7113430);