#9017322

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
prime radical	#55682	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8924518);