#10040516

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	#41424	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4170 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=gi,e^2=g,f^2=h,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, 9947712);