#8979150

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

4258 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8886346);