#844834

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=gi,d^2=h,e^2=fhi,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, 752030);