#949101

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

2010 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=fg,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 856297);