#9442896

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

4198 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9350092);