#8821950

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

4230 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8729146);