#7206226

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

1721 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)(19,30)(20,32)(21,26)(22,29)(23,28)(24,27)(25,31);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,24)(21,22)(23,31)(26,27)(28,30)(29,32);(11,15)(13,16)(17,20,23,21)(18,25,28,30)(19,22,31,24)(26,27,32,29);(1,4,6,5)(2,7,3,8)(10,14)(11,15)(17,21,23,20)(18,26,28,32)(19,22,31,24)(25,27,30,29);(1,5,6,4)(2,7,3,8)(11,15)(13,16)(17,22,23,24)(18,27,28,29)(19,20,31,21)(25,32,30,26)

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=dhi,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, 7113422);