#8840314

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

4443 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)(3,7)(4,5)(6,8)(9,10,14,15)(11,17,18,16)(12,21,23,13)(19,24,22,20)(26,30)(28,36)(31,33)(34,40);(2,5)(6,7)(10,15)(13,21)(16,17)(20,24)(26,30)(28,36)(31,33)(34,40);(1,2)(3,6)(4,5)(7,8)(11,18)(16,17)(19,22)(20,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,22)(13,20)(14,18)(15,17)(19,23)(21,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(9,12)(10,13)(11,19)(14,23)(15,21)(16,24)(17,20)(18,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(9,13)(10,12)(11,20)(14,21)(15,23)(16,22)(17,19)(18,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=ghi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8747510);