#10247659

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

4004 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=ghi,e^2=g,f^2=h,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, 10154855);