#9206621

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

3966 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)(3,10)(4,9)(5,15)(6,8)(7,13)(11,16)(12,14)(17,18)(19,22)(20,23)(21,24)(25,26)(27,32)(28,31)(29,30)(33,34,39,37)(35,42,40,36)(38,45,46,41)(43,44,47,48);(1,3,6,13)(2,7,8,10)(4,11,14,15)(5,12,16,9)(17,19)(18,22)(20,26)(21,29)(23,25)(24,30)(27,31)(28,32)(34,37)(35,40)(41,45)(43,47);(2,8)(4,14)(7,10)(11,15)(18,20)(19,25)(24,28)(27,29)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,4,6,14)(2,9,8,12)(3,11,13,15)(5,10,16,7)(17,20,23,18)(19,26,25,22)(21,24,31,28)(27,32,29,30)(33,36,39,42)(34,40,37,35)(38,44,46,48)(41,47,45,43);(1,3)(2,10)(4,15)(5,9)(6,13)(7,8)(11,14)(12,16)(17,21)(18,24)(19,27)(20,28)(22,32)(23,31)(25,29)(26,30)(33,37,39,34)(35,42,40,36)(38,45,46,41)(43,48,47,44);(1,5)(2,11)(3,12)(4,10)(6,16)(7,14)(8,15)(9,13)(17,22,23,26)(18,25,20,19)(21,30,31,32)(24,27,28,29)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=h,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, 9113817);