#8346186

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	#35703	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4352 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,19)(13,18)(14,24)(16,21)(17,23)(20,22)(25,26,32,29)(27,33,30,39)(28,37,36,34)(31,35,38,40);(2,6)(3,8)(9,11)(10,17)(12,22)(13,14)(15,21)(16,23)(18,20)(19,24)(25,27,32,30)(26,33,29,39)(28,31,36,38)(34,40,37,35);(1,3)(2,4)(5,8)(6,7)(9,12,16,13)(10,18,21,19)(11,22,23,14)(15,24,17,20)(25,28)(26,34)(27,31)(29,37)(30,38)(32,36)(33,40)(35,39);(1,3)(2,4)(5,8)(6,7)(9,13,16,12)(10,19,21,18)(11,22,23,14)(15,24,17,20)(25,29,32,26)(27,33,30,39)(28,37,36,34)(31,40,38,35);(1,4)(2,3)(5,7)(6,8)(9,14,16,22)(10,20,21,24)(11,13,23,12)(15,19,17,18)(25,30,32,27)(26,33,29,39)(28,38,36,31)(34,40,37,35);(9,15,16,17)(10,11,21,23)(12,20,13,24)(14,19,22,18)(25,31,32,38)(26,35,29,40)(27,36,30,28)(33,34,39,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=hi,e^2=hi,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, 8253382);