#9053880

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

4294 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=g,e^2=gi,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, 8961076);