#8147853

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

4615 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)(3,7)(4,6)(5,8)(14,22)(17,21)(19,20)(23,24)(25,26)(27,33)(28,30)(29,34)(31,40)(32,35)(36,39)(37,38);(1,3)(2,5)(4,8)(6,7)(9,10,15,13)(11,18,12,16)(14,17,22,21)(19,24,20,23)(25,27,32,31)(26,33,35,40)(28,37,29,36)(30,38,34,39);(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24)(25,28,32,29)(26,34,35,30)(27,36,31,37)(33,38,40,39);(1,2,4,6)(3,5,8,7)(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24)(25,29,32,28)(26,34,35,30)(27,37,31,36)(33,38,40,39);(1,2)(3,5)(4,6)(7,8)(9,13,15,10)(11,18,12,16)(14,21,22,17)(19,24,20,23)(25,30,32,34)(26,29,35,28)(27,38,31,39)(33,36,40,37);(1,3)(2,5)(4,8)(6,7)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,31,32,27)(26,33,35,40)(28,37,29,36)(30,39,34,38)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8055049);