#6624974

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573549
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×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1917 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,18)(19,27)(20,25)(21,24)(22,26)(23,28)(29,31)(30,32);(2,6)(7,8)(10,14)(11,12)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,31)(26,32);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,27)(25,31)(26,32);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,27,32,31);(9,12,13,11)(10,16,14,15)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,31,32,27)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6532170);