#7490337

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309140
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×#459	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1377 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,g^c=gi

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, 7397533);