#8374810

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	#42736	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4331 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=gh,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, 8282006);