#10464550

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

4345 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,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, 10371746);