#8509525

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

4336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=g,e^2=g,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, 8416721);