#8167439

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

4444 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8074635);