#5955560

order	512
nilpotency class	2
isoclinism	isoclinic class of #5955303
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2582 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di

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, 5862756);