#2240714

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=fi,d^2=fg,e^2=fh,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, 2147910);