#8944472

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

4134 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)(3,7)(4,8)(5,6)(9,11)(10,15)(12,16)(13,14)(17,18,23,27)(19,20,30,24)(21,25,29,32)(22,26,28,31)(34,35)(37,45)(39,41)(43,48);(1,3,6,9)(2,7,5,11)(4,10,14,12)(8,15,13,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,26)(31,32)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(2,5)(3,9)(8,13)(10,12)(21,29)(22,28)(25,32)(26,31)(33,34,38,35)(36,41,44,39)(37,42,45,40)(43,46,48,47);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,12)(11,16)(17,20,23,24)(18,19,27,30)(21,26,29,31)(22,32,28,25)(33,35,38,34)(36,39,44,41)(37,42,45,40)(43,46,48,47);(1,3)(2,7)(4,12)(5,11)(6,9)(8,16)(10,14)(13,15)(17,21,23,29)(18,25,27,32)(19,28,30,22)(20,31,24,26)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,5,6,2)(3,11,9,7)(4,13,14,8)(10,16,12,15)(17,22,23,28)(18,26,27,31)(19,29,30,21)(20,32,24,25)(33,37)(34,40)(35,42)(36,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

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

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, 8851668);