#8664393

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

4690 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571589);