#9015456

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

4060 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)(8,12)(10,16)(13,14)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30)(33,34,39,40)(35,43,44,41)(36,46,45,42)(37,38,47,48);(2,7)(4,14)(8,11)(10,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(34,40)(35,44)(37,47)(42,46);(2,7)(3,12)(4,14)(5,16)(18,20)(19,25)(21,29)(30,32)(33,35,39,44)(34,41,40,43)(36,37,45,47)(38,46,48,42);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,39,45)(34,42,40,46)(35,37,44,47)(38,43,48,41);(1,4,6,14)(2,9,7,13)(3,10,12,15)(5,11,16,8)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,37)(34,38)(35,45)(36,44)(39,47)(40,48)(41,46)(42,43);(1,5)(2,10)(3,13)(4,11)(6,16)(7,15)(8,14)(9,12)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)(33,38)(34,37)(35,46)(36,43)(39,48)(40,47)(41,45)(42,44)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=ghi,e^2=gi,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, 8922652);