#8957605

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

4423 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8864801);