#9203627

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	#35907	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4321 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=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, 9110823);