#676590

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2186 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 583786);