#6356643

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356633
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1780 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)(4,7)(8,12)(9,13)(11,14)(15,16)(18,22)(19,26)(20,23)(25,31)(28,29)(30,32);(1,2,7,12,5,8,4,6)(3,9,14,16,10,15,11,13)(17,18,23,31,21,25,20,22)(19,27,26,30,29,24,28,32);(2,8)(6,12)(9,15)(13,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,26)(25,27)(31,32);(1,3)(2,9)(4,11)(5,10)(6,13)(7,14)(8,15)(12,16);(3,10)(9,15)(11,14)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338407, #72338442, #72338448
maximal normal subgroup of	#72338407, #72338442, #72338448

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263839);