#499182

order	512
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#459
prime radical	C₂×#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1044 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406378);