#432582

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
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

930 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339778);