#8165200

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

4444 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)(3,5)(4,7)(6,8)(9,10,15,13)(11,18,17,14)(12,20,21,16)(19,22,23,24)(25,26,31,30)(27,32,38,37)(28,34,29,33)(35,40,36,39);(2,3)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,24)(21,23)(25,27,31,38)(26,32,30,37)(28,35,29,36)(33,39,34,40);(1,2,5,3)(4,6,8,7)(12,21)(16,20)(19,23)(22,24)(25,28,31,29)(26,33,30,34)(27,35,38,36)(32,39,37,40);(1,3,5,2)(4,7,8,6)(9,12,15,21)(10,16,13,20)(11,19,17,23)(14,22,18,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,4,5,8)(2,6,3,7)(9,13,15,10)(11,18,17,14)(12,16,21,20)(19,22,23,24)(25,29,31,28)(26,34,30,33)(27,36,38,35)(32,40,37,39);(9,14,15,18)(10,17,13,11)(12,22,21,24)(16,23,20,19)(25,30,31,26)(27,37,38,32)(28,34,29,33)(35,40,36,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8072396);