#7223194

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,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, 7130390);