#8348924

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

4249 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=g,f^2=g,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, 8256120);