#766212

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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

2210 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)(3,5)(4,7)(6,8)(9,10,13,15)(11,16,12,14)(18,21)(19,23)(25,26,29,32)(27,30,31,28);(2,3)(6,7)(9,10)(11,14)(12,16)(13,15)(18,21)(22,24)(25,27,29,31)(26,30,32,28);(1,2,5,3)(4,7,8,6)(9,11,13,12)(10,14,15,16)(17,18,20,21)(19,22,23,24)(25,28,29,30)(26,31,32,27);(1,3,5,2)(4,7,8,6)(9,12,13,11)(10,14,15,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,31)(29,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,16)(12,14)(13,15)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 673408);