#9206573

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

3964 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)(4,7)(5,9)(6,12)(10,15)(11,13)(14,16)(17,18)(19,25)(20,24)(21,23)(22,27)(26,32)(28,31)(29,30)(33,34,39,36)(35,42,41,38)(37,40,45,44)(43,48,47,46);(1,3,7,9)(2,8,4,5)(6,11,15,16)(10,14,12,13)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,30)(29,31)(37,45)(40,44)(43,47)(46,48);(2,4)(3,9)(10,12)(11,16)(18,23)(22,25)(24,29)(31,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,48)(45,47);(1,4,7,2)(3,5,9,8)(6,12,15,10)(11,13,16,14)(17,18)(19,22)(20,29)(21,23)(24,30)(25,27)(26,32)(28,31)(33,36,39,34)(35,42,41,38)(37,44,45,40)(43,48,47,46);(1,5)(2,9)(3,4)(6,14)(7,8)(10,11)(12,16)(13,15)(17,20)(18,24)(19,26)(21,30)(22,31)(23,29)(25,32)(27,28)(33,37,39,45)(34,40,36,44)(35,43,41,47)(38,46,42,48);(1,6,7,15)(2,10,4,12)(3,11,9,16)(5,13,8,14)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,30)(29,31)(33,38,39,42)(34,41,36,35)(37,46,45,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=gh,e^2=ghi,f^2=gh,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, 9113769);