#9938119

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

4578 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)(3,4)(5,6)(7,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,3)(2,4)(5,8)(6,7)(12,19)(15,17)(20,24)(22,23)(26,31)(28,36)(32,34)(35,37);(1,4)(2,7)(3,5)(6,8)(10,11)(15,17)(16,18)(22,23)(25,27,30,29)(26,32,31,34)(28,35,36,37)(33,39,38,40);(1,2,6,5)(3,8,7,4)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28)(26,33)(27,35)(29,37)(30,36)(31,38)(32,39)(34,40);(1,5,6,2)(3,8,7,4)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,29,30,27)(26,34,31,32)(28,35,36,37)(33,39,38,40)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=h,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, 9845315);