#8733963

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

4254 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8641159);