#8183767

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

4162 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)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(17,18,24,23)(19,27,22,25)(20,30,26,21)(28,29,31,32)(34,39)(35,36)(42,45)(43,44);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,19,24,22)(18,25,23,27)(20,31,26,28)(21,32,30,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(2,6)(7,13)(8,11)(14,15)(17,20)(18,21)(19,28)(22,31)(23,30)(24,26)(25,29)(27,32)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,48)(44,47);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,21,24,30)(18,26,23,20)(19,29,22,32)(25,31,27,28)(33,35,38,36)(34,40,39,41)(37,43,46,44)(42,47,45,48);(1,4,5,12)(2,8,6,11)(3,9,10,16)(7,14,13,15)(17,22,24,19)(18,27,23,25)(20,31,26,28)(21,32,30,29)(33,36,38,35)(34,41,39,40)(37,44,46,43)(42,48,45,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,23,24,18)(19,27,22,25)(20,30,26,21)(28,32,31,29)(33,37,38,46)(34,42,39,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8090963);