#2202508

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

1847 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=fgi,d^2=fg,e^2=fg,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, 2109704);