#8325717

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

4377 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)(11,20)(12,15)(13,23)(14,18)(16,19)(17,22)(21,24)(26,30)(31,36)(32,37)(38,40);(1,2,5,4)(3,6,7,8)(9,11)(10,17)(12,21)(13,18)(14,15)(16,22)(19,20)(23,24)(27,33)(31,36)(34,39)(38,40);(2,4)(6,8)(9,12,16,13)(10,15,19,23)(11,21,22,18)(14,20,24,17)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(3,7)(6,8)(9,13,16,12)(10,15,19,23)(11,18,22,21)(14,20,24,17)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,3)(2,6)(4,8)(5,7)(9,14)(10,18)(11,15)(12,20)(13,17)(16,24)(19,21)(22,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,4,5,2)(3,8,7,6)(9,15)(10,12)(11,14)(13,19)(16,23)(17,21)(18,20)(22,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8232913);