#9210300

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

4221 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=i,e^2=gh,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, 9117496);