#8097210

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	C₂×#550	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4395 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)(6,8)(9,10,15,14)(11,18,13,16)(12,17,20,22)(19,24,21,23)(26,28)(32,33)(34,36)(39,40);(2,5)(3,7)(12,20)(17,22)(19,21)(23,24)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,14,18)(12,19,20,21)(17,23,22,24)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24)(25,28,31,26)(27,33,29,32)(30,36,38,34)(35,40,37,39);(9,13,15,11)(10,16,14,18)(12,21,20,19)(17,23,22,24)(25,29,31,27)(26,33,28,32)(30,35,38,37)(34,39,36,40);(9,14,15,10)(11,18,13,16)(12,22,20,17)(19,24,21,23)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004406);