#8946549

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

4585 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=gi,e^2=g,f^2=g,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, 8853745);