#8107219

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	#36596	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4432 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014415);