#9027977

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

4340 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^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, 8935173);