#9050468

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

4322 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,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, 8957664);