#8152197

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

4362 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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, 8059393);