#8288391

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

4585 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8195587);