#9214757

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

4432 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)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24)(25,26,30,34)(27,31,37,39)(28,32,38,36)(29,33,40,35);(1,3)(2,6)(4,7)(5,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,27)(26,31)(28,33)(29,36)(30,37)(32,40)(34,39)(35,38);(1,2)(3,4)(5,7)(6,8)(25,28)(26,32)(27,35)(29,31)(30,38)(33,37)(34,36)(39,40);(1,4,5,6)(2,3,7,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24)(25,28)(26,32)(27,33)(29,39)(30,38)(31,40)(34,36)(35,37);(1,4)(2,3)(5,6)(7,8)(9,13,14,11)(10,15,17,18)(12,19,21,22)(16,24,20,23)(25,27)(26,31)(28,35)(29,32)(30,37)(33,38)(34,39)(36,40);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,20)(13,18)(14,17)(16,21)(19,24)(22,23)(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,d^2=h,e^2=g,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, 9121953);