#9214747

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

4470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9121943);