#8699337

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4245 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=gi,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, 8606533);