#5486512

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1865 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=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, 5393708);