#9945567

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

4346 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9852763);