#8699307

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

4240 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=gh,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, 8606503);