#8633974

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

4104 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)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(33,34,40,43)(35,44,46,41)(36,39,37,47)(38,42,48,45);(2,4)(3,10)(8,12)(11,13)(17,18)(19,24)(20,21)(22,23)(25,29)(26,30)(27,32)(28,31)(33,35,40,46)(34,41,43,44)(36,42,37,45)(38,47,48,39);(2,4)(7,9)(8,12)(15,16)(18,23)(20,29)(24,31)(26,32)(33,36,40,37)(34,39,43,47)(35,42,46,45)(38,44,48,41);(1,3,6,10)(2,7,4,9)(5,13,14,11)(8,16,12,15)(17,19)(18,24)(20,26)(21,30)(22,28)(23,31)(25,27)(29,32)(33,37,40,36)(34,39,43,47)(35,45,46,42)(38,44,48,41);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,32)(30,31)(33,38)(34,42)(35,39)(36,41)(37,44)(40,48)(43,45)(46,47);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,21,22,25)(18,20,23,29)(19,27,28,30)(24,32,31,26)(33,39)(34,37)(35,38)(36,43)(40,47)(41,45)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8541170);