#6489776

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470077
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 ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1570 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,30)(31,32);(2,5)(6,8)(9,11,14,10)(12,15,13,16)(17,19,20,27)(18,22,24,26)(21,28,29,32)(23,31,25,30);(9,12,14,13)(10,15,11,16)(18,23)(19,27)(21,29)(22,30)(24,25)(26,31);(9,12,14,13)(10,15,11,16)(19,27)(22,26)(28,32)(30,31);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(18,24)(22,26)(23,25)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396972);