#5955566

order	512
nilpotency class	2
isoclinism	isoclinic class of #5955303
automorphism	Aut(G) order 1048576
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

2198 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)(10,12)(11,15)(17,18,21,22)(19,20,23,24)(26,30)(31,32);(2,5)(6,8)(9,10,13,12)(11,16,15,14)(19,23)(20,24);(9,10,13,12)(11,14,15,16)(17,19)(18,20)(21,23)(22,24)(25,26)(27,31)(28,32)(29,30);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(17,20)(18,19)(21,24)(22,23)(25,27,29,28)(26,31,30,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,16,15,14)(17,18)(19,20)(21,22)(23,24)(25,28,29,27)(26,32,30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=g,e^2=g,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, 5862762);