#8418383

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

4721 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8325579);