#4308359

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1802 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)(9,10)(11,14)(12,16)(13,15)(17,18,20,22)(19,23,24,21)(26,27)(30,31);(2,5)(6,8)(10,11)(15,16)(25,26,29,27)(28,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,16,13,15)(17,19,20,24)(18,21,22,23)(25,27,29,26)(28,31,32,30);(9,12,14,13)(10,15,11,16)(17,18)(19,21)(20,22)(23,24)(25,28,29,32)(26,30,27,31);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,11,15)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=g,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215555);