#8599546

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	#59028	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4647 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:

(9,10,16,22)(11,20,14,17)(12,19,13,18)(15,21,24,23)(25,26,32,35)(27,36,38,33)(28,31,29,39)(30,34,40,37);(2,5)(6,8)(9,11)(10,17)(12,21)(13,23)(14,16)(15,19)(18,24)(20,22)(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)(10,18,22,19)(11,21,14,23)(15,17,24,20)(25,28,32,29)(26,31,35,39)(27,34,38,37)(30,36,40,33);(3,7)(6,8)(9,13,16,12)(10,19,22,18)(11,21,14,23)(15,17,24,20)(25,29,32,28)(26,31,35,39)(27,37,38,34)(30,36,40,33);(1,2)(3,6)(4,5)(7,8)(9,14,16,11)(10,20,22,17)(12,21,13,23)(15,18,24,19)(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)(10,21)(11,19)(12,17)(13,20)(14,18)(16,24)(22,23)(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=gh,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, 8506742);