#8183784

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

4154 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,31)(26,29)(30,32)(33,34,38,41)(35,42,37,39)(36,44,46,40)(43,47,45,48);(2,3)(7,9)(8,10)(14,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,35,38,37)(34,39,41,42)(36,43,46,45)(40,47,44,48);(5,13)(8,10)(11,16)(14,15)(18,22)(23,29)(24,27)(30,31)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,44)(42,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,21,28)(18,24,22,27)(19,25,26,32)(23,30,29,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,44)(42,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,37,38,35)(34,39,41,42)(36,45,46,43)(40,47,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=gh,e^2=gh,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, 8090980);