#9508888

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

4016 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(34,39)(35,43)(36,46)(42,47);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,18,22,23)(19,21,25,30)(20,24,27,31)(26,29,28,32)(33,34)(35,40)(36,41)(37,47)(38,39)(42,44)(43,45)(46,48);(2,7)(6,13)(8,11)(14,15)(18,23)(20,27)(21,30)(26,28)(33,35,38,43)(34,40,39,45)(36,37,46,44)(41,47,48,42);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,19,22,25)(18,21,23,30)(20,28,27,26)(24,32,31,29)(33,36,38,46)(34,41,39,48)(35,44,43,37)(40,42,45,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37,38,44)(34,42,39,47)(35,36,43,46)(40,48,45,41);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,21)(18,25)(19,23)(20,29)(22,30)(24,28)(26,31)(27,32)(33,34)(35,45)(36,41)(37,42)(38,39)(40,43)(44,47)(46,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9416084);