#10042518

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

4314 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)(3,6)(4,7)(5,10)(8,14)(9,11)(12,16)(13,15)(17,18)(19,22)(20,27)(21,23)(24,28)(25,31)(26,29)(30,32)(33,34,39,36)(35,42,37,40)(38,41,46,44)(43,48,45,47);(1,2)(3,6)(4,9)(5,8)(7,11)(10,14)(12,16)(13,15)(17,19,21,26)(18,22,23,29)(20,25,28,32)(24,30,27,31)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(2,3)(4,11)(8,10)(12,13)(18,23)(22,29)(24,27)(30,31)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(1,3,6,2)(4,9,11,7)(5,10,14,8)(12,16,13,15)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,28)(31,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,4)(2,7)(3,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,37,39,35)(34,40,36,42)(38,43,46,45)(41,48,44,47);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,20,21,28)(18,24,23,27)(19,25,26,32)(22,30,29,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9949714);