#674113

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2002 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=fgi,d^2=fgh,e^2=gh,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, 581309);