#8864595

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

4638 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:

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

Polycyclic presentation:

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

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, 8771791);