#8108437

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

4544 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015633);