#8174322

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

4585 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8081518);