#8127107

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

4596 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=g,e^2=g,f^2=h,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, 8034303);