#10247304

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

4158 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(18,22)(20,28)(23,25)(26,29)(33,34)(35,36)(37,45)(38,40)(39,47)(41,43)(42,46)(44,48);(2,3)(7,9)(8,10)(15,16)(17,18,21,22)(19,25,27,23)(20,24,28,30)(26,32,29,31)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,46)(47,48);(17,18,21,22)(19,23,27,25)(20,24,28,30)(26,31,29,32)(33,34,38,40)(35,41,43,36)(37,39,47,45)(42,46,48,44);(1,3,6,2)(4,7,12,9)(5,10,14,8)(11,15,13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,34)(35,41)(36,43)(37,39)(38,40)(42,46)(44,48)(45,47);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,19,21,27)(18,23,22,25)(20,29,28,26)(24,32,30,31)(33,36,38,41)(34,35,40,43)(37,46,47,44)(39,48,45,42);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,20,21,28)(18,24,22,30)(19,26,27,29)(23,31,25,32)(33,37)(34,39)(35,42)(36,44)(38,47)(40,45)(41,46)(43,48)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=ghi,e^2=h,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, 10154500);