#9015289

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

4447 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)(3,7)(4,5)(6,8)(9,10,15,21)(11,16,23,24)(12,18,22,17)(13,19,20,14)(25,26,30,32)(27,33,35,29)(28,31,38,36)(34,39,40,37);(2,5)(6,7)(9,11,15,23)(10,16,21,24)(12,20,22,13)(14,17,19,18)(25,27)(26,29)(28,34)(30,35)(31,37)(32,33)(36,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,22)(10,17,21,18)(11,13,23,20)(14,24,19,16)(25,26,30,32)(27,29,35,33)(28,36,38,31)(34,39,40,37);(1,3,4,8)(2,6,5,7)(9,12)(10,18)(11,20)(13,23)(14,16)(15,22)(17,21)(19,24)(25,26)(27,29)(28,36)(30,32)(31,38)(33,35)(34,39)(37,40);(9,13,15,20)(10,19,21,14)(11,22,23,12)(16,17,24,18)(25,28,30,38)(26,31,32,36)(27,34,35,40)(29,37,33,39);(9,14,15,19)(10,20,21,13)(11,17,23,18)(12,16,22,24)(25,29)(26,27)(28,37)(30,33)(31,34)(32,35)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8922485);