#9057323

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

4304 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)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(17,18,23,25)(19,24,20,29)(21,26,31,30)(22,28,32,27)(35,41)(39,42)(43,47)(46,48);(2,7)(4,10)(8,15)(12,14)(17,19,23,20)(18,24,25,29)(21,28,31,27)(22,26,32,30)(33,34,37,38)(35,39,41,42)(36,40,44,45)(43,46,47,48);(3,11)(4,10)(5,9)(8,15)(18,25)(19,20)(21,31)(22,32)(34,38)(35,41)(36,44)(46,48);(1,3,6,11)(2,8,7,15)(4,14,10,12)(5,13,9,16)(17,20,23,19)(18,24,25,29)(21,28,31,27)(22,30,32,26)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,47)(45,48);(1,4,6,10)(2,9,7,5)(3,12,11,14)(8,13,15,16)(17,21)(18,26)(19,28)(20,27)(22,29)(23,31)(24,32)(25,30)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,45)(46,47);(1,5,6,9)(2,10,7,4)(3,13,11,16)(8,12,15,14)(17,22,23,32)(18,27,25,28)(19,26,20,30)(21,29,31,24)(33,36,37,44)(34,40,38,45)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gh,e^2=g,f^2=gi,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, 8964519);