#4650108

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

2656 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)(3,4)(5,6)(7,8)(10,11)(12,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32);(2,6)(3,8)(9,10,13,11)(12,14,16,15)(17,19,22,21)(18,23,24,20)(26,27)(30,31);(2,6)(4,7)(12,16)(14,15)(18,24)(19,21)(25,26,29,27)(28,31,32,30);(1,3,5,8)(2,7,6,4)(9,11,13,10)(12,15,16,14)(17,20,22,23)(18,19,24,21)(25,27,29,26)(28,31,32,30);(1,4,5,7)(2,8,6,3)(9,12,13,16)(10,14,11,15)(17,21,22,19)(18,23,24,20)(25,28,29,32)(26,30,27,31)

Polycyclic presentation:

2^9|b^2=gi,c^2=fhi,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, 4557304);