#2555260

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2051 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fhi,c^2=fghi,d^2=f,e^2=fghi,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, 2462456);