#2159628

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1921 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=fgh,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2066824);