#8165213

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

4769 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=g,e^2=i,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, 8072409);