#8183742

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

4115 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,31)(29,30)(33,34,38,40)(35,39,42,36)(37,45,46,41)(43,44,48,47);(2,7)(3,10)(9,14)(11,16)(17,19,23,20)(18,24,25,26)(21,28,32,30)(22,29,27,31)(33,35)(34,39)(36,40)(37,43)(38,42)(41,47)(44,45)(46,48);(5,13)(9,14)(11,16)(12,15)(18,25)(22,27)(24,26)(29,31)(34,40)(35,42)(41,45)(43,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,20,23,19)(18,26,25,24)(21,30,32,28)(22,31,27,29)(33,36,38,39)(34,35,40,42)(37,44,46,47)(41,43,45,48);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,21)(18,27)(19,28)(20,30)(22,25)(23,32)(24,31)(26,29)(33,37,38,46)(34,41,40,45)(35,43,42,48)(36,44,39,47);(1,5,6,13)(2,9,7,14)(3,11,10,16)(4,12,8,15)(17,22,23,27)(18,21,25,32)(19,29,20,31)(24,28,26,30)(33,35)(34,39)(36,40)(37,43)(38,42)(41,47)(44,45)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=h,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, 8090938);