#8201966

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

4249 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8109162);