#9017186

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

4146 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)(9,15)(11,13)(17,18)(19,24)(20,25)(21,30)(22,23)(26,32)(27,31)(28,29)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(1,2,6,7)(3,8,10,5)(4,9,12,15)(11,16,13,14)(17,18)(19,24)(20,28)(21,26)(22,23)(25,29)(27,31)(30,32)(34,38)(39,41)(40,44)(47,48);(3,10)(4,12)(5,8)(9,15)(18,23)(19,27)(20,29)(21,32)(33,35,37,43)(34,39,38,41)(36,45,46,42)(40,48,44,47);(1,3)(2,8)(4,13)(5,7)(6,10)(9,14)(11,12)(15,16)(17,19,22,27)(18,24,23,31)(20,30,29,26)(21,28,32,25)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,48)(45,46);(1,4,6,12)(2,9,7,15)(3,11,10,13)(5,14,8,16)(17,20)(18,25)(19,26)(21,31)(22,29)(23,28)(24,32)(27,30)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(1,5,6,8)(2,3,7,10)(4,14,12,16)(9,11,15,13)(17,21,22,32)(18,26,23,30)(19,28,27,25)(20,31,29,24)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45)

Polycyclic presentation:

2^9|b^2=ghi,c^2=hi,d^2=i,e^2=ghi,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, 8924382);