#9174328

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

4594 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9081524);