#8197810

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

4487 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=g,e^2=g,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

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, 8105006);