#474282

order	512
nilpotency class	3
isoclinism	isoclinic class of #459486
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

816 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381478);