#9029774

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

4395 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)(3,7)(4,8)(5,6)(9,10,14,18)(11,17,21,15)(12,13,23,16)(19,24,22,20)(25,26)(27,33)(28,38)(29,35)(30,36)(31,32)(34,39)(37,40);(2,6)(3,8)(9,11,14,21)(10,15,18,17)(12,19,23,22)(13,20,16,24)(25,27,32,29)(26,33,31,35)(28,30,39,37)(34,40,38,36);(1,3)(2,4)(5,8)(6,7)(9,12,14,23)(10,16,18,13)(11,19,21,22)(15,24,17,20)(25,28,32,39)(26,34,31,38)(27,37,29,30)(33,36,35,40);(1,4)(2,3)(5,7)(6,8)(9,12)(10,16)(11,19)(13,18)(14,23)(15,24)(17,20)(21,22)(25,29,32,27)(26,35,31,33)(28,37,39,30)(34,36,38,40);(1,2)(3,4)(5,6)(7,8)(9,11)(10,17)(12,22)(13,20)(14,21)(15,18)(16,24)(19,23)(25,30,32,37)(26,36,31,40)(27,28,29,39)(33,38,35,34);(1,4)(2,3)(5,7)(6,8)(9,13,14,16)(10,12,18,23)(11,20,21,24)(15,19,17,22)(25,31,32,26)(27,35,29,33)(28,34,39,38)(30,40,37,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=gh,e^2=gh,f^2=gi,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, 8936970);