#8110827

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

4484 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)(12,20)(16,21)(18,24)(22,23)(25,26,30,34)(27,28,32,31)(29,37,38,33)(35,39,40,36);(3,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,32)(26,31,34,28)(29,35,38,40)(33,39,37,36);(1,2)(3,6)(4,5)(7,8)(10,14)(15,17)(16,21)(22,23)(25,28,30,31)(26,32,34,27)(29,36,38,39)(33,40,37,35);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,28)(26,27)(29,36)(30,31)(32,34)(33,35)(37,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,28)(29,35)(30,32)(31,34)(33,36)(37,39)(38,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8018023);