#4650457

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 2097152
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

1826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,3)(4,8)(9,10)(11,13)(12,15)(14,16)(19,23)(21,24)(26,30)(27,33)(28,35)(38,40);(1,2,5,3)(4,6,8,7)(9,11,14,15)(10,12,16,13)(17,18,20,22)(19,21,23,24)(26,30)(27,33)(32,37)(34,39);(4,8)(6,7)(9,12,14,13)(10,15,16,11)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,5,2)(4,7,8,6)(9,13,14,12)(10,15,16,11)(17,19)(18,21)(20,23)(22,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,4)(2,6)(3,7)(5,8)(9,10)(11,12)(13,15)(14,16)(17,19)(18,21)(20,23)(22,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=fhi,d^2=fg,e^2=g,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, 4557653);