#8733978

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

4085 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)(17,18,22,25)(19,20,27,23)(21,24,31,32)(26,28,29,30)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,19,22,27)(18,23,25,20)(21,26,31,29)(24,30,32,28)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(3,11)(7,8)(12,16)(13,14)(17,19,22,27)(18,23,25,20)(21,26,31,29)(24,30,32,28)(33,35,39,37)(34,41,36,40)(38,43,46,45)(42,48,44,47);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,19)(18,20)(21,29)(22,27)(23,25)(24,30)(26,31)(28,32)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,20)(18,19)(21,30)(22,23)(24,29)(25,27)(26,32)(28,31)(33,37,39,35)(34,40,36,41)(38,45,46,43)(42,47,44,48);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,21)(18,24)(19,26)(20,28)(22,31)(23,30)(25,32)(27,29)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=gh,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, 8641174);