#6357757

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

1944 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=h,e^2=i,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, 6264953);