#8358545

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

4302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,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, 8265741);