#8898304

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

4324 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=gh,e^2=h,f^2=g,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, 8805500);