#7291212

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234333
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1775 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7198408);