#9206619

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
prime radical	#46723	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4010 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,28)(21,22)(23,27)(25,29)(26,30)(31,32)(33,34,39,37)(35,42,36,40)(38,41,46,45)(43,48,44,47);(1,3)(2,6)(4,10)(5,13)(7,9)(8,14)(11,15)(12,16)(18,22)(23,24)(25,28)(30,31)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(2,4)(3,9)(8,12)(11,13)(17,19,21,27)(18,23,22,24)(20,26,29,32)(25,30,28,31)(33,34,39,37)(35,40,36,42)(38,41,46,45)(43,47,44,48);(1,4,7,2)(3,10,9,6)(5,8,15,12)(11,16,13,14)(17,19)(18,24)(20,26)(21,27)(22,23)(25,31)(28,30)(29,32)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,5)(2,8)(3,11)(4,12)(6,16)(7,15)(9,13)(10,14)(17,18)(19,23)(20,28)(21,22)(24,27)(25,29)(26,31)(30,32)(33,37,39,34)(35,42,36,40)(38,41,46,45)(43,47,44,48);(1,6,7,10)(2,9,4,3)(5,14,15,16)(8,11,12,13)(17,20,21,29)(18,25,22,28)(19,26,27,32)(23,30,24,31)(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=ghi,c^2=g,d^2=gh,e^2=ghi,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, 9113815);