#6353830

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353803
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
prime radical	#61950	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1589 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261026);