#6184925

order	512
nilpotency class	2
isoclinism	isoclinic class of #6132643
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

1294 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=fg,d^2=fgh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef

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, 6092121);