#8463259

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

4411 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=h,e^2=hi,f^2=i,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, 8370455);