#8999416

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

4430 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=ghi,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, 8906612);