#8534577

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

4704 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8441773);