#9174305

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

4502 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:

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

Polycyclic presentation:

2^9|b^2=hi,e^2=g,f^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, 9081501);