#8158924

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

4323 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,f^2=h,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, 8066120);