#8599434

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
prime radical	#36530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4477 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506630);