#8826219

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

4146 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)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(17,18,22,24)(19,23,25,32)(20,28,30,21)(26,27,29,31)(34,36)(35,41)(40,44)(42,47);(2,7)(4,9)(5,14)(11,16)(17,19)(18,23)(20,26)(21,31)(22,25)(24,32)(27,28)(29,30)(33,34,38,36)(35,39,41,43)(37,44,45,40)(42,48,47,46);(2,7)(3,10)(8,13)(11,16)(19,25)(23,32)(26,29)(27,31)(35,41)(39,43)(42,47)(46,48);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,35,38,41)(34,39,36,43)(37,42,45,47)(40,46,44,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20)(18,21)(19,26)(22,30)(23,31)(24,28)(25,29)(27,32)(33,36,38,34)(35,39,41,43)(37,44,45,40)(42,46,47,48);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,21)(18,20)(19,27)(22,28)(23,29)(24,30)(25,31)(26,32)(33,37,38,45)(34,40,36,44)(35,42,41,47)(39,46,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8733415);