#7231168

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200082
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
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 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)(3,7)(4,8)(5,6)(9,10,13,14,12,15,11,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31);(2,6)(3,4)(9,11,12,13)(10,14,15,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,26)(30,31);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,15)(13,16);(2,6)(7,8)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32);(1,4,5,3)(2,7,6,8)(17,21,22,20)(18,24,25,30)(19,27,28,29)(23,32,26,31)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7138364);