#2241609

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

1458 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=fgi,d^2=f,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, 2148805);