#9210326

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	#35719	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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)(3,7)(4,11)(5,8)(6,9)(10,15)(12,16)(13,14)(17,18)(19,20)(21,26)(22,29)(23,25)(24,28)(27,31)(30,32)(33,34,39,41)(35,40,42,38)(36,45,46,37)(43,44,48,47);(1,3,6,4)(2,7,9,11)(5,10,14,12)(8,15,13,16)(17,19,23,28)(18,24,25,20)(21,31,32,22)(26,29,30,27)(33,35)(34,40)(36,43)(37,47)(38,41)(39,42)(44,45)(46,48);(3,4)(7,11)(10,12)(15,16)(18,25)(20,24)(26,30)(27,29)(34,41)(35,42)(36,46)(44,47);(1,4,6,3)(2,7,9,11)(5,12,14,10)(8,15,13,16)(17,20)(18,19)(21,27)(22,30)(23,24)(25,28)(26,31)(29,32)(33,36,39,46)(34,37,41,45)(35,43,42,48)(38,44,40,47);(1,2)(3,7)(4,11)(5,13)(6,9)(8,14)(10,16)(12,15)(17,21,23,32)(18,26,25,30)(19,22,28,31)(20,29,24,27)(33,37)(34,36)(35,44)(38,43)(39,45)(40,48)(41,46)(42,47);(1,5,6,14)(2,8,9,13)(3,10,4,12)(7,15,11,16)(17,22)(18,27)(19,21)(20,30)(23,31)(24,26)(25,29)(28,32)(33,38)(34,42)(35,41)(36,44)(37,48)(39,40)(43,45)(46,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=i,e^2=g,f^2=ghi,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, 9117522);