#8718448

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

4442 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)(9,10,15,19)(11,16,20,23)(12,17,21,22)(13,14,18,24)(25,26,30,29)(27,32,34,31)(28,36,37,33)(35,38,40,39);(2,5)(6,7)(9,11,15,20)(10,16,19,23)(12,13,21,18)(14,22,24,17)(27,34)(31,32)(35,40)(38,39);(1,2)(3,6)(4,5)(7,8)(11,20)(12,21)(16,23)(17,22)(25,27,30,34)(26,31,29,32)(28,35,37,40)(33,38,36,39);(1,3)(2,6)(4,8)(5,7)(9,12,15,21)(10,17,19,22)(11,13,20,18)(14,23,24,16)(25,27)(26,32)(28,35)(29,31)(30,34)(33,39)(36,38)(37,40);(9,13)(10,14)(11,21)(12,20)(15,18)(16,22)(17,23)(19,24)(25,28,30,37)(26,33,29,36)(27,35,34,40)(31,38,32,39);(9,14)(10,18)(11,22)(12,16)(13,19)(15,24)(17,20)(21,23)(25,29,30,26)(27,32,34,31)(28,33,37,36)(35,38,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8625644);