#9909774

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

4303 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)(9,10,14,15)(11,13,17,22)(12,16,20,23)(18,21,24,19);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,26,31,28)(27,33,29,32)(30,34,38,36)(35,40,37,39);(1,3)(2,4)(5,8)(6,7)(10,15)(11,17)(16,23)(18,24)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,28,31,26)(27,33,29,32)(30,36,38,34)(35,40,37,39);(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29,31,27)(26,33,28,32)(30,35,38,37)(34,39,36,40);(9,13)(10,11)(12,19)(14,22)(15,17)(16,24)(18,23)(20,21)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=ghi,e^2=ghi,f^2=ghi,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, 9816970);