#8696783

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	#36266	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4330 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603979);