#10042529

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4347 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=gi,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, 9949725);