#8152202

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

4346 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)(9,10,15,13)(11,19,22,16)(12,23,21,17)(14,24,20,18)(26,30)(27,33)(28,36)(38,40);(3,7)(6,8)(9,11)(10,16)(12,20)(13,19)(14,21)(15,22)(17,18)(23,24)(25,26,29,30)(27,34,33,31)(28,32,36,37)(35,40,39,38);(1,2)(3,6)(4,5)(7,8)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12,15,21)(10,17,13,23)(11,20,22,14)(16,18,19,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40);(1,2)(3,6)(4,5)(7,8)(9,13,15,10)(11,19,22,16)(12,17,21,23)(14,24,20,18)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40);(9,14,15,20)(10,18,13,24)(11,21,22,12)(16,17,19,23)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8059398);