#9053870

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4417 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961066);