#10042488

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

4122 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,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, 9949684);