#8930384

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

4297 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)(3,8)(4,13)(5,10)(6,16)(7,9)(11,15)(12,14)(19,25)(23,30)(26,28)(31,32)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,47)(44,48);(1,3,7,4)(2,8,9,13)(5,11,16,14)(6,12,10,15)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(2,9)(3,4)(6,10)(11,14)(18,22)(19,25)(24,27)(26,28)(33,34,38,41)(35,39,43,36)(37,45,46,40)(42,44,48,47);(1,4,7,3)(2,8,9,13)(5,11,16,14)(6,15,10,12)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,5)(2,10)(3,11)(4,14)(6,9)(7,16)(8,15)(12,13)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,6,7,10)(2,5,9,16)(3,12,4,15)(8,11,13,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,37,38,46)(34,40,41,45)(35,42,43,48)(36,44,39,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837580);