#10055526

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

4129 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)(3,8,11,14)(4,9,10,5)(12,15,16,13)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,34)(35,39)(36,41)(37,42)(38,40)(43,46)(44,47)(45,48);(2,7)(3,11)(4,10)(13,15)(17,18,21,24)(19,22,25,29)(20,23,28,32)(26,30,27,31)(33,35,38,42)(34,39,40,37)(36,44,45,43)(41,47,48,46);(2,7)(3,11)(4,10)(13,15)(19,25)(22,29)(26,27)(30,31)(34,40)(35,42)(36,45)(46,47);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,15)(10,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,35)(34,37)(36,43)(38,42)(39,40)(41,47)(44,45)(46,48);(1,4)(2,9)(3,12)(5,7)(6,10)(8,15)(11,16)(13,14)(17,19)(18,22)(20,27)(21,25)(23,31)(24,29)(26,28)(30,32)(33,36,38,45)(34,41,40,48)(35,43,42,44)(37,47,39,46);(1,5,6,9)(2,10,7,4)(3,13,11,15)(8,16,14,12)(17,20,21,28)(18,23,24,32)(19,26,25,27)(22,30,29,31)(33,37)(34,35)(36,46)(38,39)(40,42)(41,44)(43,48)(45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,e^2=i,f^2=g,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, 9962722);