#7206898

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

1602 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18)(19,28)(20,24)(21,27)(22,26)(23,25)(29,32)(30,31);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19)(18,24)(20,22)(21,30)(23,29)(25,31)(26,28)(27,32);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(18,25)(21,23)(24,31)(26,27)(28,32)(29,30);(2,6)(3,7)(9,11,13,12)(10,16,14,15)(17,20,22,19)(18,24,26,28)(21,29,23,30)(25,31,27,32);(3,7)(5,8)(9,12,13,11)(10,16,14,15)(17,21,22,23)(18,25,26,27)(19,29,20,30)(24,32,28,31)

Polycyclic presentation:

2^9|c^2=fi,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=dfh,e^b=ehi,e^c=ei,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, 7114094);