#9863358

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

4491 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=ghi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9770554);