#9017373

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	#55159	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4073 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)(3,11)(4,9)(5,15)(6,7)(8,13)(10,16)(12,14)(17,18,23,27)(19,24,22,32)(20,21,26,25)(28,30,29,31);(2,7)(4,14)(8,11)(10,15)(17,19,23,22)(18,24,27,32)(20,29,26,28)(21,31,25,30)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(1,3,6,13)(2,8,7,11)(4,15,14,10)(5,9,16,12)(17,20,23,26)(18,25,27,21)(19,28,22,29)(24,31,32,30)(33,35,39,46)(34,40,43,44)(36,37,42,45)(38,47,48,41);(1,3)(2,8)(4,15)(5,12)(6,13)(7,11)(9,16)(10,14)(17,20)(18,21)(19,29)(22,28)(23,26)(24,31)(25,27)(30,32)(33,36,39,42)(34,38,43,48)(35,45,46,37)(40,41,44,47);(1,4,6,14)(2,9,7,12)(3,10,13,15)(5,11,16,8)(17,21)(18,26)(19,30)(20,27)(22,31)(23,25)(24,29)(28,32)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,5)(2,10)(3,12)(4,11)(6,16)(7,15)(8,14)(9,13)(17,22,23,19)(18,24,27,32)(20,28,26,29)(21,31,25,30)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=ghi,e^2=hi,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, 8924569);