#8375610

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	#37662	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4629 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8282806);