#9935798

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

4463 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)(5,8)(6,7)(9,10)(11,13)(12,16)(14,15)(17,22)(18,21)(19,24)(20,23)(25,26)(27,29)(28,31)(30,36)(32,34)(33,38)(35,39)(37,40);(1,3,7,8)(2,4,6,5)(10,15)(11,17)(16,23)(18,24)(25,27)(26,32)(28,29)(30,35)(31,34)(33,39)(36,37)(38,40);(1,2,7,6)(3,5,8,4)(10,15)(11,17)(16,23)(18,24)(26,28)(27,34)(33,36)(35,40);(1,4,7,5)(2,8,6,3)(9,11,14,17)(10,13,15,22)(12,18,20,24)(16,21,23,19)(25,28,31,26)(27,29,34,32)(30,36,38,33)(35,37,40,39);(1,5,7,4)(2,8,6,3)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29)(26,27)(28,34)(30,37)(31,32)(33,35)(36,40)(38,39);(1,6,7,2)(3,5,8,4)(9,13)(10,17)(11,15)(12,19)(14,22)(16,18)(20,21)(23,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=h,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, 9842994);