#8402678

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

4525 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)(9,10,15,20)(11,16,12,17)(13,18,23,22)(14,19,24,21)(25,26,31,29)(27,28,35,32)(30,33,38,37)(34,36,40,39);(1,3,4,8)(2,5,6,7)(11,12)(13,23)(16,17)(18,22)(25,27)(26,32)(28,29)(30,34)(31,35)(33,39)(36,37)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,12)(10,16,20,17)(13,21,23,19)(14,22,24,18)(25,26,31,29)(27,32,35,28)(30,37,38,33)(34,36,40,39);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,17,20,16)(13,21,23,19)(14,22,24,18)(25,28)(26,27)(29,35)(30,36)(31,32)(33,34)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,13)(10,18)(11,21)(12,19)(14,16)(15,23)(17,24)(20,22)(25,29,31,26)(27,32,35,28)(30,37,38,33)(34,39,40,36);(9,14,15,24)(10,19,20,21)(11,18,12,22)(13,16,23,17)(25,30,31,38)(26,33,29,37)(27,34,35,40)(28,36,32,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=hi,e^2=gh,f^2=gi,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, 8309874);