#8096945

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

4555 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004141);