#8692571

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

4450 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8599767);