#8159060

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	#37473	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066256);