#9169365

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

4136 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,29)(21,26)(22,25)(23,24)(28,31)(30,32)(33,34,39,44)(35,45,47,40)(36,46,48,41)(37,42,43,38);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,21)(20,28)(22,32)(23,27)(24,26)(25,31)(29,30)(33,34,39,44)(35,45,47,40)(36,41,48,46)(37,38,43,42);(3,10)(6,7)(11,16)(14,15)(17,20,24,22)(18,25,23,29)(19,28,26,32)(21,31,27,30)(33,35,39,47)(34,40,44,45)(36,43,48,37)(38,46,42,41);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,21,24,27)(18,26,23,19)(20,30,22,31)(25,28,29,32)(33,36,39,48)(34,41,44,46)(35,37,47,43)(38,40,42,45);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,22,24,20)(18,25,23,29)(19,28,26,32)(21,30,27,31)(33,37)(34,42)(35,36)(38,44)(39,43)(40,41)(45,46)(47,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,23,24,18)(19,27,26,21)(20,29,22,25)(28,30,32,31)(33,38,39,42)(34,43,44,37)(35,46,47,41)(36,40,48,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

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, 9076561);