#616940

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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

1830 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fhi,c^2=g,d^2=f,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 524136);