#9210378

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

4215 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)(3,5)(4,8)(6,7)(9,10,15,19)(11,13,18,16)(12,14,22,17)(20,23,24,21)(25,26,30,33)(27,34,29,31)(28,36,38,32)(35,39,37,40);(2,7)(3,8)(9,11,15,18)(10,16,19,13)(12,20,22,24)(14,21,17,23)(27,29)(31,34)(35,37)(39,40);(1,3,6,8)(2,4,7,5)(9,12,15,22)(10,17,19,14)(11,20,18,24)(13,21,16,23)(25,27,30,29)(26,31,33,34)(28,37,38,35)(32,40,36,39);(1,4,6,5)(2,3,7,8)(9,12)(10,14)(11,20)(13,23)(15,22)(16,21)(17,19)(18,24)(25,28,30,38)(26,32,33,36)(27,35,29,37)(31,39,34,40);(1,2)(3,4)(5,8)(6,7)(9,13)(10,18)(11,19)(12,21)(14,20)(15,16)(17,24)(22,23)(25,29,30,27)(26,31,33,34)(28,35,38,37)(32,40,36,39);(1,5,6,4)(2,8,7,3)(9,14)(10,12)(11,21)(13,24)(15,17)(16,20)(18,23)(19,22)(25,26)(27,31)(28,32)(29,34)(30,33)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,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, 9117574);