#8943991

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

4238 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)(4,12)(5,9)(6,7)(8,14)(11,15)(13,16)(17,18,23,27)(19,20,30,24)(21,25,29,32)(22,26,28,31)(33,34)(35,41)(36,45)(37,39)(38,42)(40,43)(44,47)(46,48);(1,3,7,9)(2,5,6,10)(4,11,14,16)(8,15,12,13)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,26)(31,32)(33,35,40,44)(34,41,43,47)(36,46,37,42)(38,45,48,39);(2,6)(5,10)(8,12)(13,15)(21,29)(22,28)(25,32)(26,31)(33,36,40,37)(34,39,43,45)(35,42,44,46)(38,41,48,47);(1,4)(2,8)(3,11)(5,15)(6,12)(7,14)(9,16)(10,13)(17,20,23,24)(18,19,27,30)(21,26,29,31)(22,32,28,25)(33,37,40,36)(34,39,43,45)(35,42,44,46)(38,47,48,41);(1,5,7,10)(2,9,6,3)(4,13,14,15)(8,11,12,16)(17,21,23,29)(18,25,27,32)(19,28,30,22)(20,31,24,26)(33,38)(34,42)(35,39)(36,47)(37,41)(40,48)(43,46)(44,45);(1,6,7,2)(3,10,9,5)(4,12,14,8)(11,13,16,15)(17,22,23,28)(18,26,27,31)(19,29,30,21)(20,32,24,25)(33,39)(34,36)(35,38)(37,43)(40,45)(41,46)(42,47)(44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=gh,e^2=g,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, 8851187);