#8156813

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

4687 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)(11,13)(12,20)(14,15)(16,18)(17,22)(19,23)(21,24)(25,26)(27,32)(28,37)(29,39)(30,33)(31,34)(35,36)(38,40);(3,7)(6,8)(10,15)(11,18)(17,20)(19,24)(27,34)(31,32)(35,38)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,27,30,34)(26,31,33,32)(28,35,39,38)(29,36,37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,27)(26,32)(28,38)(29,36)(30,34)(31,33)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28)(26,29)(27,35)(30,39)(31,36)(32,40)(33,37)(34,38);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,29)(26,28)(27,36)(30,37)(31,35)(32,38)(33,39)(34,40)

Polycyclic presentation:

2^9|c^2=gi,e^2=g,f^2=g,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, 8064009);