#8720331

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

4480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627527);