#8388978

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

4496 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8296174);