#8639623

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	#34829	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4380 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gi,d^2=g,f^2=gi,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, 8546819);