#8183780

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

4123 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)(17,18,23,22)(19,25,20,24)(21,29,30,26)(27,31,28,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,47)(42,45)(46,48);(2,7)(3,10)(9,14)(11,16)(17,19,23,20)(18,24,22,25)(21,27,30,28)(26,31,29,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,45)(43,44)(47,48);(5,13)(9,14)(11,16)(12,15)(18,22)(24,25)(26,29)(31,32)(34,38)(39,45)(40,43)(46,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,20,23,19)(18,25,22,24)(21,28,30,27)(26,32,29,31)(33,35,37,42)(34,39,38,45)(36,41,44,48)(40,46,43,47);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,36,37,44)(34,40,38,43)(35,41,42,48)(39,46,45,47);(1,5,6,13)(2,9,7,14)(3,11,10,16)(4,12,8,15)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,45)(43,44)(47,48)

Polycyclic presentation:

2^9|a^2=hi,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, 8090976);