#8968124

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

4486 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)(3,4)(5,6)(7,8)(9,10)(11,12)(13,21)(14,15)(16,19)(17,22)(18,23)(20,24)(26,31)(28,32)(33,39)(36,40);(2,6)(4,7)(10,15)(12,16)(17,21)(20,23)(25,26,30,31)(27,32,35,28)(29,33,38,39)(34,40,37,36);(1,2,5,6)(3,7,8,4)(10,15)(12,16)(13,22)(18,24)(25,27,30,35)(26,32,31,28)(29,37,38,34)(33,36,39,40);(1,3)(2,7)(4,6)(5,8)(9,11,14,19)(10,16,15,12)(13,18,22,24)(17,23,21,20)(25,28)(26,27)(29,36)(30,32)(31,35)(33,34)(37,39)(38,40);(1,4,5,7)(2,3,6,8)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,40)(36,39)(37,38);(1,3)(2,7)(4,6)(5,8)(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=hi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=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, 8875320);