#6357728

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357716
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1949 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264924);