#8421708

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

4596 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8328904);