#6348996

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348967
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2883 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,23,27)(19,25,30,31)(20,24,29,32)(21,28,22,26);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,30)(18,24,27,32)(20,21,29,22)(25,26,31,28);(17,20)(18,25)(19,21)(22,30)(23,29)(24,26)(27,31)(28,32);(17,21,23,22)(18,26,27,28)(19,29,30,20)(24,31,32,25);(2,7)(6,15)(8,12)(9,16)(18,27)(24,32)(25,31)(26,28)

Polycyclic presentation:

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

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, 6256192);