#8648926

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	#42679	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=hi,e^2=gh,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, 8556122);