#8267602

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

4615 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:

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

Polycyclic presentation:

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

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, 8174798);