#8472381

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

4509 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8379577);