#8826257

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

4336 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(17,18,22,24)(19,23,25,32)(20,28,30,21)(26,27,29,31)(33,34)(35,36)(37,45)(38,39)(40,41)(42,46)(43,47)(44,48);(2,7)(5,9)(8,10)(13,16)(17,19)(18,23)(20,26)(21,31)(22,25)(24,32)(27,28)(29,30)(34,39)(36,40)(37,46)(43,48);(3,11)(8,10)(12,14)(13,16)(19,25)(23,32)(26,29)(27,31)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,3)(2,8)(4,14)(5,13)(6,11)(7,10)(9,16)(12,15)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,35)(34,40)(36,39)(37,43)(38,41)(42,47)(44,45)(46,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,20)(18,21)(19,26)(22,30)(23,31)(24,28)(25,29)(27,32)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,5,6,9)(2,4,7,15)(3,13,11,16)(8,14,10,12)(17,21)(18,20)(19,27)(22,28)(23,29)(24,30)(25,31)(26,32)(33,37,38,46)(34,42,39,45)(35,43,41,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=h,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, 8733453);