#5593658

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1967 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)(3,5,7,8)(9,10,13,14)(11,12,15,16)(17,18)(19,21)(20,22)(23,24)(25,26,28,29)(27,30,32,31);(9,11,13,15)(10,12,14,16)(17,19,20,23)(18,21,22,24)(26,29)(30,31);(10,14)(11,15)(17,18,20,22)(19,21,23,24)(25,26)(27,31)(28,29)(30,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,30)(28,32)(29,31);(1,3,4,7)(2,5,6,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,21)(20,23)(22,24)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=gh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=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, 5500854);