#9206678

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	#48564	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4384 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)(11,15)(12,13)(14,17)(16,20)(18,19)(21,24)(22,23)(25,26)(27,29)(28,37)(30,33)(31,34)(32,36)(35,38)(39,40);(3,7)(5,8)(9,11)(10,15)(12,18)(13,19)(14,20)(16,17)(21,22)(23,24)(25,27)(26,32)(28,35)(29,34)(30,40)(31,36)(33,38)(37,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,20)(10,15,17,16)(12,18,22,21)(13,19,23,24)(25,28,31,30)(26,33,34,37)(27,35,36,40)(29,39,32,38);(9,11)(10,16)(12,21)(13,19)(14,20)(15,17)(18,22)(23,24)(25,29,31,32)(26,27,34,36)(28,38,30,39)(33,40,37,35);(1,3)(2,5)(4,7)(6,8)(9,12,14,22)(10,13,17,23)(11,18,20,21)(15,19,16,24)(25,30,31,28)(26,33,34,37)(27,35,36,40)(29,38,32,39);(1,2)(3,5)(4,6)(7,8)(9,13)(10,12)(11,19)(14,23)(15,18)(16,21)(17,22)(20,24)(25,27)(26,32)(28,35)(29,34)(30,40)(31,36)(33,38)(37,39)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=g,e^2=gi,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, 9113874);