#9026915

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	#48100	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4110 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=ghi,d^2=gh,f^2=i,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, 8934111);