#9015311

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

4274 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)(3,11,12,8)(4,9,15,14)(5,13,16,10)(17,18,23,26)(19,27,29,24)(20,30,31,21)(22,25,28,32)(33,34,39,45)(35,40,47,48)(36,42,46,41)(37,43,44,38);(2,7)(4,15)(8,11)(10,13)(17,19,23,29)(18,24,26,27)(20,25,31,32)(21,28,30,22)(33,35,39,47)(34,40,45,48)(36,44,46,37)(38,41,43,42);(2,7)(3,12)(4,15)(5,16)(20,31)(21,30)(22,28)(25,32)(33,36,39,46)(34,41,45,42)(35,37,47,44)(38,48,43,40);(1,3)(2,8)(4,10)(5,14)(6,12)(7,11)(9,16)(13,15)(17,20,23,31)(18,21,26,30)(19,25,29,32)(22,24,28,27)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,45)(43,48);(1,4)(2,9)(3,13)(5,8)(6,15)(7,14)(10,12)(11,16)(17,21)(18,20)(19,28)(22,29)(23,30)(24,25)(26,31)(27,32)(33,37,39,44)(34,43,45,38)(35,46,47,36)(40,41,48,42);(1,5)(2,10)(3,14)(4,8)(6,16)(7,13)(9,12)(11,15)(17,22,23,28)(18,25,26,32)(19,21,29,30)(20,24,31,27)(33,38,39,43)(34,44,45,37)(35,41,47,42)(36,40,46,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=ghi,e^2=i,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, 8922507);