#669470

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2222 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)(10,11)(14,15)(17,18)(19,23)(20,21)(22,24)(25,26)(27,31)(28,32)(29,30);(2,4)(6,7)(9,10,13,11)(12,15,16,14)(18,21)(22,23)(26,30)(27,32);(9,10,13,11)(12,14,16,15)(17,18)(19,22)(20,21)(23,24)(25,27,29,32)(26,31,30,28);(1,3)(2,6)(4,7)(5,8)(9,11,13,10)(12,15,16,14)(17,19,20,24)(18,22,21,23)(25,28,29,31)(26,32,30,27);(1,4,5,2)(3,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,30)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=fgi,d^2=gh,e^2=fgh,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, 576666);