#8891767

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

4627 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8798963);