#8157185

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

4436 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=g,e^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, 8064381);