#9935802

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

4431 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)(3,4)(5,8)(6,7)(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,22)(23,24)(25,26)(27,30)(28,29)(31,35)(32,34)(33,39)(36,40)(37,38);(1,3,7,8)(2,4,6,5)(9,11,15,14)(10,16,19,20)(12,23,18,21)(13,24,17,22)(25,27)(26,32)(28,36)(29,38)(30,35)(31,34)(33,40)(37,39);(1,2,7,6)(3,5,8,4)(25,28,31,39)(26,33,35,29)(27,36,34,37)(30,38,32,40);(1,4,7,5)(2,8,6,3)(9,12,15,18)(10,17,19,13)(11,21,14,23)(16,24,20,22)(25,28)(26,29)(27,37)(30,38)(31,39)(32,40)(33,35)(34,36);(1,5,7,4)(2,8,6,3)(9,13,15,17)(10,18,19,12)(11,22,14,24)(16,23,20,21)(25,29)(26,28)(27,38)(30,37)(31,33)(32,36)(34,40)(35,39);(1,6,7,2)(3,5,8,4)(9,14,15,11)(10,16,19,20)(12,23,18,21)(13,22,17,24)(25,30)(26,34)(27,35)(28,38)(29,36)(31,32)(33,37)(39,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 9842998);