#9159952

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

4200 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)(3,4)(5,13)(6,7)(8,10)(9,15)(11,14)(12,16)(17,18)(19,23)(20,25)(21,28)(22,24)(26,32)(27,30)(29,31)(33,34,39,44)(35,45,48,40)(36,41,47,42)(37,43,46,38);(2,7)(4,8)(9,13)(12,14)(17,19,22,27)(18,23,24,30)(20,26,31,28)(21,25,32,29)(33,35)(34,40)(36,46)(37,47)(38,42)(39,48)(41,43)(44,45);(2,7)(3,10)(9,13)(11,16)(18,24)(19,27)(25,29)(26,28)(33,36,39,47)(34,41,44,42)(35,46,48,37)(38,40,43,45);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,18)(19,23)(20,29)(21,26)(22,24)(25,31)(27,30)(28,32)(33,36)(34,42)(35,46)(37,48)(38,40)(39,47)(41,44)(43,45);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,27)(30,32)(33,37,39,46)(34,43,44,38)(35,47,48,36)(40,41,45,42);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21)(18,26)(19,25)(20,30)(22,32)(23,31)(24,28)(27,29)(33,38,39,43)(34,37,44,46)(35,42,48,41)(36,40,47,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=hi,e^2=g,f^2=g,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, 9067148);