#6357829

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357820
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
prime radical	#3508	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1890 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265025);