#6358013

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357957
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

1851 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)(3,11)(4,8)(5,9)(6,16)(7,10)(12,14)(13,15)(17,18,24,19)(20,30,31,32)(21,25,28,29)(22,26,23,27);(1,3,6,8,7,13,5,12)(2,4,9,11,10,14,16,15)(17,19,22,26,24,18,23,27)(20,28,25,30,31,21,29,32);(17,20,23,29,24,31,22,25)(18,21,26,30,19,28,27,32);(1,4,7,14)(2,8,10,12)(3,9,13,16)(5,15,6,11)(17,21,24,28)(18,25,19,29)(20,27,31,26)(22,32,23,30);(1,5,7,6)(2,9,10,16)(3,12,13,8)(4,11,14,15)(17,22,24,23)(18,26,19,27)(20,25,31,29)(21,30,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265209);