#8992966

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

4204 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8900162);