#9019988

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

4487 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=g,f^2=gi,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, 8927184);