#10247281

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

4012 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,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, 10154477);