#10480809

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

4142 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=i,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, 10388005);