#8118831

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

4476 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)(3,5)(4,7)(6,8)(9,10)(11,19)(12,22)(13,21)(14,15)(16,20)(17,23)(18,24)(25,26)(27,35)(28,32)(29,30)(31,34)(33,40)(36,39)(37,38);(2,3)(6,7)(10,15)(11,20)(17,22)(18,21)(25,27,31,28)(26,32,34,35)(29,36,40,38)(30,37,33,39);(1,2,5,3)(4,6,8,7)(10,15)(12,23)(16,19)(18,21)(25,28,31,27)(26,32,34,35)(29,38,40,36)(30,37,33,39);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,23)(22,24)(25,29)(26,33)(27,36)(28,38)(30,34)(31,40)(32,39)(35,37);(1,4,5,8)(2,6,3,7)(9,12,14,23)(10,17,15,22)(11,21,20,18)(13,19,24,16)(25,30,31,33)(26,29,34,40)(27,37,28,39)(32,36,35,38);(9,13)(10,18)(11,17)(12,19)(14,24)(15,21)(16,23)(20,22)(25,29)(26,33)(27,36)(28,38)(30,34)(31,40)(32,39)(35,37)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=i,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026027);