#8844545

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

4369 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:

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

Polycyclic presentation:

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

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, 8751741);