#5236501

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5143697);