#615615

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

2186 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=gi,d^2=g,e^2=f,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, 522811);