#8127173

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

4472 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)(11,15)(12,19)(13,21)(14,17)(16,22)(18,20)(23,24)(26,31)(28,34)(32,40)(35,38);(3,7)(6,8)(9,11,14,19)(10,15,17,12)(13,18,22,24)(16,23,21,20)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10,14,17)(11,12,19,15)(13,16,22,21)(18,20,24,23)(25,27,30,37)(26,32,31,40)(28,38,34,35)(29,39,33,36);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,24)(25,28,30,34)(26,33,31,29)(27,35,37,38)(32,39,40,36);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,29,30,33)(26,34,31,28)(27,36,37,39)(32,38,40,35);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034369);