#8527580

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

4740 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)(11,19)(12,22)(13,18)(14,16)(15,21)(17,23)(20,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40);(3,7)(5,8)(9,11,14,21)(10,15,16,19)(12,20,23,18)(13,17,24,22)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,39)(37,40);(1,2)(3,5)(4,6)(7,8)(10,16)(12,23)(15,19)(18,20)(25,27,29,35)(26,31,33,30)(28,34,37,40)(32,39,36,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23)(25,26)(27,31)(28,36)(29,33)(30,35)(32,37)(34,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,20)(13,19)(14,23)(15,24)(16,22)(18,21)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(9,13,14,24)(10,18,16,20)(11,17,21,22)(12,19,23,15)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40)

Polycyclic presentation:

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

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, 8434776);