#8366642

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

4118 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,15)(13,16)(33,34,39,37)(35,40,36,42)(38,41,46,45)(43,47,44,48);(1,3,5,11)(2,6,7,9)(4,13,14,10)(8,16,12,15)(17,18,22,23)(19,24,28,30)(20,27,31,25)(21,26,32,29)(34,37)(40,42)(41,45)(47,48);(2,7)(6,9)(8,12)(15,16)(18,23)(21,32)(24,30)(25,27)(33,35,39,36)(34,40,37,42)(38,44,46,43)(41,48,45,47);(1,3)(2,6)(4,13)(5,11)(7,9)(8,16)(10,14)(12,15)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,32)(26,31)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,4)(2,8)(3,10)(5,14)(6,15)(7,12)(9,16)(11,13)(17,20,22,31)(18,25,23,27)(19,26,28,29)(21,30,32,24)(33,37,39,34)(35,42,36,40)(38,45,46,41)(43,48,44,47);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,21)(18,26)(19,27)(20,30)(22,32)(23,29)(24,31)(25,28)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8273838);