#9668370

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

4183 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)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,18)(19,21)(20,29)(22,30)(23,25)(24,28)(26,27)(31,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,23,28)(18,24,25,21)(20,26,30,32)(22,27,29,31)(33,35)(34,38)(36,44)(37,43)(39,47)(40,41)(42,45)(46,48);(2,6)(3,10)(8,12)(11,16)(18,25)(20,30)(21,24)(26,32)(33,35,37,43)(34,38,40,41)(36,42,45,44)(39,46,48,47);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,20,23,30)(18,22,25,29)(19,26,28,32)(21,31,24,27)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,21,23,24)(18,19,25,28)(20,27,30,31)(22,32,29,26)(33,36)(34,39)(35,42)(37,45)(38,46)(40,48)(41,47)(43,44);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22)(18,20)(19,27)(21,32)(23,29)(24,26)(25,30)(28,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575566);