#8630866

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

4446 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)(3,4)(5,8)(6,7)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(26,31)(27,34)(28,36)(38,39);(1,3)(2,4)(5,6)(7,8)(9,11,14,19)(10,15,13,17)(12,18,21,24)(16,22,20,23)(25,26)(27,29)(28,37)(30,31)(32,36)(33,34)(35,38)(39,40);(1,4)(2,5)(3,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,27,30,34)(26,29,31,33)(28,35,36,40)(32,39,37,38);(1,3,6,5)(2,4,7,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,28)(26,32)(27,35)(29,39)(30,36)(31,37)(33,38)(34,40);(1,5,6,3)(2,4,7,8)(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,27)(26,33)(28,35)(29,31)(30,34)(32,38)(36,40)(37,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,29,30,33)(26,27,31,34)(28,38,36,39)(32,40,37,35)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8538062);