#10086356

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
prime radical	#42393	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4192 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,d^2=gi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9993552);