#9184011

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	#35352	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4513 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=g,f^2=g,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, 9091207);