#8533633

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

4548 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)(9,10,16,20)(11,21,24,17)(12,18,13,14)(15,19,23,22)(25,26)(27,34)(28,38)(29,33)(30,31)(32,37)(35,36)(39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,19)(13,22)(14,23)(15,18)(16,24)(20,21)(26,31)(28,33)(32,34)(35,39);(1,3)(2,5)(4,8)(6,7)(9,12,16,13)(10,18,20,14)(11,19,24,22)(15,21,23,17)(25,27,30,37)(26,32,31,34)(28,35,33,39)(29,36,38,40);(1,3)(2,5)(4,8)(6,7)(9,13,16,12)(10,18,20,14)(11,22,24,19)(15,21,23,17)(25,28)(26,29)(27,35)(30,33)(31,38)(32,36)(34,40)(37,39);(1,2)(3,5)(4,6)(7,8)(9,14,16,18)(10,12,20,13)(11,23,24,15)(17,19,21,22)(25,27)(26,32)(28,35)(29,36)(30,37)(31,34)(33,39)(38,40);(9,15,16,23)(10,19,20,22)(11,18,24,14)(12,21,13,17)(25,29)(26,33)(27,36)(28,31)(30,38)(32,39)(34,35)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,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,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440829);