#8772558

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

4443 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,e^2=gh,f^2=h,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, 8679754);