#8898902

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

4460 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)(9,10,15,17)(11,16,14,21)(12,18,22,23)(13,19,24,20)(26,27)(29,37)(31,33)(35,40);(1,2,5,8)(3,6,4,7)(9,11,15,14)(10,16,17,21)(12,20,22,19)(13,23,24,18)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(1,3,5,4)(2,6,8,7)(10,17)(11,14)(18,23)(19,20)(25,26,30,27)(28,31,36,33)(29,34,37,32)(35,39,40,38);(9,12)(10,18)(11,20)(13,16)(14,19)(15,22)(17,23)(21,24)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,2)(3,6)(4,7)(5,8)(9,13)(10,19)(11,18)(12,21)(14,23)(15,24)(16,22)(17,20)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4,5,3)(2,7,8,6)(9,14,15,11)(10,16,17,21)(12,19,22,20)(13,23,24,18)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806098);