#10464554

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10371750);