#9935789

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

4295 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)(9,10,15,16)(11,18,13,12)(14,22,23,17)(19,20,21,24)(27,33)(31,37)(34,39)(38,40);(1,3,7,8)(2,4,6,5)(9,11,15,13)(10,12,16,18)(14,19,23,21)(17,20,22,24)(26,30)(27,33)(32,35)(34,39);(1,2,7,6)(3,5,8,4)(10,16)(12,18)(17,22)(20,24)(25,26)(27,31)(28,32)(29,30)(33,37)(34,38)(35,36)(39,40);(1,4,7,5)(2,8,6,3)(9,12)(10,11)(13,16)(14,20)(15,18)(17,19)(21,22)(23,24)(25,27)(26,31)(28,34)(29,33)(30,37)(32,38)(35,40)(36,39);(1,5,7,4)(2,8,6,3)(9,13,15,11)(10,12,16,18)(14,19,23,21)(17,24,22,20)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,6,7,2)(3,5,8,4)(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,22)(18,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=gh,e^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, 9842985);