#463900

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

809 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)(5,6,11,14,10,15,9,25)(7,16,26,28,18,32,19,12)(8,20,23,30,24,13,22,29)(17,33,34,27,21,31,35,36);(5,7,9,19,10,18,11,26)(6,12,14,16,15,28,25,32)(8,21,22,34,24,17,23,35)(13,27,29,31,20,36,30,33);(5,8,9,22,10,24,11,23)(6,13,14,29,15,20,25,30)(7,17,19,35,18,21,26,34)(12,27,16,31,28,36,32,33);(2,4)(7,18)(8,22,24,23)(12,28)(13,29,20,30)(16,32)(17,34,21,35)(19,26)(27,33,36,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371096);