#8850601

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

4250 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)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(17,18,21,23)(19,25,27,22)(20,28,30,24)(26,31,29,32)(33,34)(35,38)(36,44)(37,41)(39,43)(40,46)(42,48)(45,47);(2,6)(7,9)(8,12)(15,16)(17,19,21,27)(18,22,23,25)(20,29,30,26)(24,32,28,31)(33,35,37,43)(34,38,41,39)(36,45,46,42)(40,48,44,47);(3,10)(7,9)(11,13)(15,16)(18,23)(22,25)(24,28)(31,32)(33,34,37,41)(35,39,43,38)(36,40,46,44)(42,48,45,47);(1,3,5,10)(2,7,6,9)(4,13,14,11)(8,16,12,15)(17,18)(19,22)(20,28)(21,23)(24,30)(25,27)(26,32)(29,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(1,4,5,14)(2,8,6,12)(3,11,10,13)(7,15,9,16)(17,20,21,30)(18,24,23,28)(19,26,27,29)(22,31,25,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,48)(44,47)(45,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,31)(28,32)(29,30)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,44)(43,45)

Polycyclic presentation:

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

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, 8757797);