#8943934

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

4227 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)(3,9)(4,8)(5,13)(6,7)(10,14)(11,15)(12,16)(17,18,23,25)(19,26,20,24)(21,22,32,31)(27,30,29,28)(34,39)(36,40)(41,47)(44,48);(2,7)(8,9)(10,13)(15,16)(17,18,23,25)(19,26,20,24)(21,31,32,22)(27,28,29,30)(33,34,38,39)(35,40,43,36)(37,41,46,47)(42,48,45,44);(1,3,6,4)(2,8,7,9)(5,12,14,11)(10,16,13,15)(17,19,23,20)(18,24,25,26)(21,29,32,27)(22,30,31,28)(33,35,38,43)(34,40,39,36)(37,45,46,42)(41,44,47,48);(1,4,6,3)(2,9,7,8)(5,12,14,11)(10,16,13,15)(17,20,23,19)(18,24,25,26)(21,27,32,29)(22,30,31,28)(33,36)(34,35)(37,44)(38,40)(39,43)(41,42)(45,47)(46,48);(1,2)(3,9)(4,8)(5,10)(6,7)(11,16)(12,15)(13,14)(17,21)(18,22)(19,27)(20,29)(23,32)(24,28)(25,31)(26,30)(33,35)(34,40)(36,39)(37,42)(38,43)(41,48)(44,47)(45,46);(1,5,6,14)(2,10,7,13)(3,11,4,12)(8,15,9,16)(17,22)(18,21)(19,28)(20,30)(23,31)(24,27)(25,32)(26,29)(33,37,38,46)(34,41,39,47)(35,42,43,45)(36,44,40,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=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, 8851130);