#8889413

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

4094 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)(3,10,9,5)(4,12,13,8)(11,15,14,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,34)(35,40)(36,44)(37,47)(38,39)(41,46)(42,45)(43,48);(2,7)(4,13)(5,10)(11,14)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(34,39)(40,48)(41,44)(42,47);(2,7)(3,9)(8,12)(11,14)(18,22)(23,25)(24,28)(30,31)(33,35,38,43)(34,40,39,48)(36,45,46,37)(41,47,44,42);(1,3,6,9)(2,5,7,10)(4,14,13,11)(8,16,12,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36,38,46)(34,41,39,44)(35,37,43,45)(40,42,48,47);(1,4)(2,8)(3,11)(5,15)(6,13)(7,12)(9,14)(10,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,37)(34,42)(35,36)(38,45)(39,47)(40,41)(43,46)(44,48);(1,5)(2,9)(3,7)(4,15)(6,10)(8,14)(11,12)(13,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796609);