#10247170

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

4162 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)(3,4)(5,13)(6,7)(8,15)(9,10)(11,14)(12,16)(18,22)(20,28)(23,25)(26,29)(33,34)(35,45)(36,41)(37,47)(38,42)(39,43)(40,44)(46,48);(1,3)(2,4)(5,11)(6,9)(7,10)(8,12)(13,14)(15,16)(17,18,21,22)(19,25,27,23)(20,24,28,30)(26,32,29,31)(33,35)(34,39)(36,46)(37,38)(40,43)(41,42)(44,45)(47,48);(2,7)(3,9)(8,13)(11,16)(17,18,21,22)(19,23,27,25)(20,24,28,30)(26,31,29,32)(33,36,40,37)(34,41,44,47)(35,46,43,38)(39,42,45,48);(1,2)(3,10)(4,9)(5,8)(6,7)(11,14)(12,16)(13,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,37,40,36)(34,41,44,47)(35,46,43,38)(39,48,45,42);(1,4,6,10)(2,3,7,9)(5,14,15,12)(8,16,13,11)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,28)(30,31)(33,38,40,46)(34,42,44,48)(35,37,43,36)(39,41,45,47);(1,5)(2,8)(3,11)(4,12)(6,15)(7,13)(9,16)(10,14)(17,20,21,28)(18,24,22,30)(19,26,27,29)(23,31,25,32)(33,39,40,45)(34,43,44,35)(36,42,37,48)(38,47,46,41)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,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, 10154366);