#8648982

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

4408 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,d^2=hi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8556178);