#7787784

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

4447 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7694980);