#4616412

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2813 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)(11,16)(12,15)(13,14)(18,21)(19,20)(25,26,31,30)(27,28,32,29);(3,8)(5,7)(9,11,14,12)(10,15,13,16)(17,18,22,21)(19,23,20,24)(25,27)(26,29)(28,30)(31,32);(1,3)(2,5)(4,8)(6,7)(9,11,14,12)(10,16,13,15)(17,19,22,20)(18,23,21,24)(25,28,31,29)(26,32,30,27);(1,3)(2,5)(4,8)(6,7)(9,12,14,11)(10,16,13,15)(17,20,22,19)(18,23,21,24)(25,29,31,28)(26,32,30,27);(1,2)(3,5)(4,6)(7,8)(9,13,14,10)(11,16,12,15)(17,21,22,18)(19,23,20,24)(25,30,31,26)(27,28,32,29)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4523608);