#8155052

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

4656 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=g,f^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^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, 8062248);