#8550948

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

4362 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458144);