#8115774

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

4719 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)(6,8)(9,10,14,15)(11,19,20,16)(12,21,22,17)(13,23,24,18)(25,26,31,30)(27,34,37,32)(28,33,29,38)(35,40,36,39);(1,2)(3,6)(4,5)(7,8)(10,15)(11,20)(17,21)(18,23)(25,27)(26,32)(28,35)(29,36)(30,34)(31,37)(33,39)(38,40);(2,5)(6,8)(10,15)(12,22)(16,19)(18,23)(25,28,31,29)(26,33,30,38)(27,35,37,36)(32,39,34,40);(9,11,14,20)(10,16,15,19)(12,18,22,23)(13,21,24,17)(25,29,31,28)(26,33,30,38)(27,36,37,35)(32,39,34,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,22)(10,17,15,21)(11,18,20,23)(13,19,24,16)(25,30,31,26)(27,34,37,32)(28,33,29,38)(35,39,36,40);(1,3)(2,6)(4,7)(5,8)(9,13)(10,18)(11,21)(12,19)(14,24)(15,23)(16,22)(17,20)(25,27)(26,32)(28,35)(29,36)(30,34)(31,37)(33,39)(38,40)

Polycyclic presentation:

2^9|a^2=g,c^2=gh,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,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, 8022970);