#8850570

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

4120 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=ghi,e^2=g,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, 8757766);