#8159104

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

4295 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)(3,5)(4,7)(6,8)(9,10,15,13)(11,18,14,16)(12,20,21,17)(19,23,22,24)(25,26)(27,33)(28,35)(29,37)(30,32)(31,38)(34,36)(39,40);(2,3)(6,7)(9,11,15,14)(10,16,13,18)(12,19,21,22)(17,23,20,24)(25,26,30,32)(27,28,35,33)(29,31,38,37)(34,36,40,39);(1,2,5,3)(4,6,8,7)(12,21)(17,20)(19,22)(23,24)(25,27)(26,28)(29,34)(30,35)(31,36)(32,33)(37,39)(38,40);(1,3,5,2)(4,7,8,6)(9,12,15,21)(10,17,13,20)(11,19,14,22)(16,23,18,24)(25,28,30,33)(26,27,32,35)(29,36,38,39)(31,34,37,40);(1,4)(2,6)(3,7)(5,8)(9,13,15,10)(11,18,14,16)(12,17,21,20)(19,23,22,24)(25,27)(26,28)(29,34)(30,35)(31,36)(32,33)(37,39)(38,40);(9,14,15,11)(10,18,13,16)(12,22,21,19)(17,24,20,23)(25,29)(26,31)(27,34)(28,36)(30,38)(32,37)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066300);