#9863140

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

4488 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9770336);