#10216707

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4301 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=hi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10123903);