#9713756

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

4391 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=h,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, 9620952);