#8976412

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

4697 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)(4,6)(5,8)(10,15)(11,17)(16,22)(18,23)(25,26,30,33)(27,32,36,31)(28,29,39,37)(34,40,38,35);(1,2,4,6)(3,5,8,7)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,26,30,33)(27,32,36,31)(28,37,39,29)(34,35,38,40);(1,2)(3,5)(4,6)(7,8)(11,17)(12,19)(13,21)(16,22)(25,27,30,36)(26,31,33,32)(28,38,39,34)(29,40,37,35);(1,3,4,8)(2,5,6,7)(9,11,14,17)(10,12,15,19)(13,18,21,23)(16,20,22,24)(25,27)(26,32)(28,38)(29,35)(30,36)(31,33)(34,39)(37,40);(9,12)(10,11)(13,20)(14,19)(15,17)(16,18)(21,24)(22,23)(25,28)(26,29)(27,34)(30,39)(31,35)(32,40)(33,37)(36,38);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,28)(27,35)(30,37)(31,34)(32,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=g,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, 8883608);