#413506

order	512
nilpotency class	3
isoclinism	isoclinic class of #393899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320702);