#8436554

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

4466 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8343750);