#9206520

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

4091 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)(3,8)(4,9)(5,13)(6,7)(10,15)(11,12)(14,16)(17,18)(19,24)(20,23)(21,25)(22,27)(26,32)(28,31)(29,30)(33,34,38,39)(35,41,42,36)(37,45,47,40)(43,44,48,46);(2,7)(4,8)(10,13)(12,14)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,34,38,39)(35,36,42,41)(37,40,47,45)(43,44,48,46);(2,7)(3,9)(10,13)(11,16)(20,28)(23,31)(26,29)(30,32)(34,39)(35,42)(40,45)(43,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,14)(12,13)(15,16)(17,20,21,28)(18,23,25,31)(19,26,27,29)(22,30,24,32)(33,35,38,42)(34,36,39,41)(37,43,47,48)(40,44,45,46);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,19)(18,24)(20,29)(21,27)(22,25)(23,30)(26,28)(31,32)(33,36)(34,35)(37,46)(38,41)(39,42)(40,48)(43,45)(44,47);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,31)(29,32)(33,37,38,47)(34,40,39,45)(35,43,42,48)(36,44,41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gh,e^2=hi,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, 9113716);