#8798609

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

4268 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8705805);