#4632832

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,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, 4540028);