#8325629

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	#40599	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4301 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^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, 8232825);