#9160827

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	#58216	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,20)(13,14)(16,22)(17,19)(18,24)(21,23)(25,26,31,35)(27,28,36,32)(29,37,39,33)(30,34,38,40);(1,2,5,6)(3,7,8,4)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,27)(26,32)(28,35)(29,34)(30,37)(31,36)(33,38)(39,40);(1,3)(2,4)(5,8)(6,7)(10,14)(15,17)(16,20)(23,24)(25,28,31,32)(26,27,35,36)(29,38,39,30)(33,40,37,34);(1,4)(2,3)(5,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,22)(21,23)(25,29,31,39)(26,33,35,37)(27,34,36,40)(28,30,32,38);(1,2)(3,4)(5,6)(7,8)(9,11,13,19)(10,15,14,17)(12,21,22,18)(16,24,20,23)(25,30)(26,34)(27,33)(28,29)(31,38)(32,39)(35,40)(36,37);(1,4)(2,3)(5,7)(6,8)(9,12,13,22)(10,16,14,20)(11,18,19,21)(15,23,17,24)(25,27)(26,32)(28,35)(29,34)(30,37)(31,36)(33,38)(39,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9068023);