#8726454

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4242 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633650);