#6355957

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355925
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1561 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=i,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263153);