#8975337

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

4449 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)(3,8)(4,7)(5,6)(9,10)(11,19)(12,17)(13,23)(14,15)(16,20)(18,24)(21,22)(25,26,30,33)(27,31,34,38)(28,29,32,37)(35,36,39,40);(3,4)(7,8)(10,15)(12,22)(16,19)(18,23)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40);(1,3)(2,7)(4,6)(5,8)(10,15)(11,20)(12,22)(13,24)(27,34)(28,32)(29,37)(31,38);(3,4)(7,8)(9,11)(10,16)(12,23)(13,21)(14,20)(15,19)(17,24)(18,22)(25,28)(26,29)(27,35)(30,32)(31,36)(33,37)(34,39)(38,40);(1,4,6,3)(2,7,5,8)(9,12,14,22)(10,17,15,21)(11,18,20,23)(13,19,24,16)(25,29)(26,32)(27,36)(28,33)(30,37)(31,39)(34,40)(35,38);(1,5,6,2)(3,7,4,8)(9,13)(10,18)(11,21)(12,19)(14,24)(15,23)(16,22)(17,20)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8882533);