#9210286

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

4007 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)(3,10,14,7)(4,9,15,13)(5,12,16,11)(17,18,23,22)(19,24,27,28)(20,29,30,25)(21,26,31,32)(33,34)(35,44)(36,41)(37,42)(38,40)(39,43)(45,46)(47,48);(1,3,6,14)(2,7,8,10)(4,12,15,11)(5,13,16,9)(17,19)(18,24)(20,21)(22,28)(23,27)(25,32)(26,29)(30,31)(33,35,39,38)(34,40,43,44)(36,45,42,47)(37,46,41,48);(2,8)(4,15)(7,10)(11,12)(19,27)(20,30)(24,28)(25,29)(33,36,39,42)(34,41,43,37)(35,45,38,47)(40,48,44,46);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,20,23,30)(18,25,22,29)(19,21,27,31)(24,32,28,26)(33,36)(34,41)(35,45)(37,43)(38,47)(39,42)(40,48)(44,46);(1,3)(2,10)(4,11)(5,9)(6,14)(7,8)(12,15)(13,16)(17,21)(18,26)(19,20)(22,32)(23,31)(24,29)(25,28)(27,30)(33,37,39,41)(34,42,43,36)(35,46,38,48)(40,47,44,45);(1,5)(2,11)(3,13)(4,7)(6,16)(8,12)(9,14)(10,15)(17,22,23,18)(19,28,27,24)(20,29,30,25)(21,26,31,32)(33,38,39,35)(34,40,43,44)(36,47,42,45)(37,46,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=i,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=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, 9117482);