#5987684

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 1048576
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1876 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5894880);