#10216556

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	#40504	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4295 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)(12,19)(13,16)(15,20)(17,21)(18,24)(22,23)(25,26)(27,33)(28,38)(29,35)(30,34)(31,32)(36,40)(37,39);(1,3,6,5)(2,7,4,8)(11,17)(14,21)(18,23)(22,24)(25,27,32,29)(26,33,31,35)(28,37,30,36)(34,40,38,39);(9,10,13,16)(11,14,17,21)(12,15,20,19)(18,22,23,24)(25,28,32,30)(26,34,31,38)(27,36,29,37)(33,39,35,40);(2,4)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,20)(23,24)(25,29,32,27)(26,35,31,33)(28,36,30,37)(34,39,38,40);(1,4,6,2)(3,8,5,7)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,30,32,28)(26,34,31,38)(27,37,29,36)(33,39,35,40);(1,5,6,3)(2,7,4,8)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,23)(21,24)(25,31,32,26)(27,35,29,33)(28,38,30,34)(36,40,37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=i,e^2=gi,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, 10123752);