#8957574

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

4459 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)(3,7)(4,5)(6,8)(9,10,15,12)(11,16,18,14)(13,17,21,20)(19,23,24,22)(25,26)(27,33)(28,29)(30,38)(31,32)(34,39)(35,37)(36,40);(2,5)(6,7)(9,11,15,18)(10,16,12,14)(13,19,21,24)(17,23,20,22)(26,32)(29,33)(34,38)(37,40);(1,2)(3,6)(4,5)(7,8)(11,18)(13,21)(14,16)(17,20)(25,27,31,28)(26,29,32,33)(30,36,39,35)(34,40,38,37);(1,3,4,8)(2,6,5,7)(9,12,15,10)(11,14,18,16)(13,17,21,20)(19,23,24,22)(25,28,31,27)(26,33,32,29)(30,36,39,35)(34,40,38,37);(9,13)(10,17)(11,19)(12,20)(14,22)(15,21)(16,23)(18,24)(25,29,31,33)(26,28,32,27)(30,37,39,40)(34,36,38,35);(9,14)(10,11)(12,18)(13,22)(15,16)(17,19)(20,24)(21,23)(25,30,31,39)(26,34,32,38)(27,35,28,36)(29,37,33,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=hi,e^2=h,f^2=h,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, 8864770);