#826417

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1782 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fgh,c^2=gh,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 733613);