#425933

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
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

940 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)(7,9)(8,20,23,32)(10,14)(11,16)(12,24,31,25)(13,27)(15,29)(17,33,34,30)(18,26)(19,28)(21,36,35,22);(1,2)(3,4)(5,7,10,18,11,19,29,13)(6,9,14,26,16,28,15,27)(8,21,24,34,23,35,25,17)(12,30,20,22,31,33,32,36);(2,4)(5,8,10,25,11,23,29,24)(6,12,14,32,16,31,15,20)(7,17,18,35,19,34,13,21)(9,22,26,30,28,36,27,33);(5,9)(6,13)(7,14)(8,22)(10,26)(11,28)(12,21)(15,19)(16,18)(17,32)(20,34)(23,36)(24,33)(25,30)(27,29)(31,35)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^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=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi

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, 333129);