#8496833

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

4652 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8404029);