#2555266

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2790 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=fghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2462462);