#7233745

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(18,20)(19,26)(21,23)(24,31)(27,32)(28,30);(2,5)(3,7)(10,14)(11,12)(17,18,22,20)(19,27,28,24)(21,29,23,25)(26,31,30,32);(9,11,13,12)(10,15,14,16)(17,19,22,28)(18,24,20,27)(21,26,23,30)(25,31,29,32);(1,3,4,7)(2,6,5,8)(10,14)(15,16)(17,20,22,18)(19,27,28,24)(21,25,23,29)(26,31,30,32);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,25,20,29)(19,26,28,30)(24,31,27,32)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=fhi,e^2=h,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, 7140941);