#10247319

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

4180 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154515);