#9712848

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

4383 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9620044);