#6020071

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1364 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,6,5)(3,4,8,7)(9,10,13,12)(11,15,16,14)(19,25)(20,27)(22,28)(23,30)(33,34,38,42)(35,39,40,36)(37,41,46,48)(43,47,45,44);(2,5)(3,8)(10,12)(14,15)(19,25)(22,28)(26,31)(29,32)(33,35,38,40)(34,36,42,39)(37,45,46,43)(41,47,48,44);(1,3,6,8)(2,7,5,4)(9,11)(10,14)(12,15)(13,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,35)(34,39)(36,42)(37,43)(38,40)(41,47)(44,48)(45,46);(1,4)(2,3)(5,8)(6,7)(9,10,13,12)(11,14,16,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,45,48,43);(1,5,6,2)(3,4,8,7)(9,12,13,10)(11,15,16,14)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)(33,37,38,46)(34,41,42,48)(35,43,40,45)(36,44,39,47)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=fgi,d^2=gh,e^2=f,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, 5927267);