#8276432

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

4801 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,e^2=g,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, 8183628);