#6122162

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1827 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,16)(13,15)(18,22)(20,23)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(2,5)(7,8)(9,11,13,16)(10,12,15,14)(17,18,21,22)(19,20,24,23)(26,30)(28,36)(31,33)(34,40);(1,3,6,4)(2,7,5,8)(9,11,13,16)(10,14,15,12)(17,19,21,24)(18,23,22,20)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,4,6,3)(2,8,5,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,5,6,2)(3,8,4,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,19)(21,23)(22,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fg,c^2=hi,d^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6029358);