#8201990

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

4297 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)(18,24)(19,21)(20,28)(27,32)(33,34)(35,36)(37,45)(38,40)(39,47)(41,43)(42,48)(44,46);(2,4)(7,9)(8,12)(14,15)(17,18,23,24)(19,25,21,31)(20,29,28,22)(26,27,30,32)(33,35,38,43)(34,36,40,41)(37,46,47,42)(39,48,45,44);(5,13)(8,12)(10,16)(14,15)(17,19,23,21)(18,25,24,31)(20,26,28,30)(22,32,29,27)(33,34,38,40)(35,41,43,36)(37,39,47,45)(42,48,46,44);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20)(18,22)(19,26)(21,30)(23,28)(24,29)(25,32)(27,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,46)(47,48);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21,23,19)(18,25,24,31)(20,30,28,26)(22,32,29,27)(33,37)(34,39)(35,42)(36,44)(38,47)(40,45)(41,48)(43,46);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,20)(19,27)(21,32)(23,29)(24,28)(25,30)(26,31)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,e^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, 8109186);