#8325644

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	#41337	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4191 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8232840);