#9136912

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

4150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=h,e^2=gh,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^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9044108);