#446378

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

703 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)(5,6,9,16)(7,12)(8,21,23,30)(10,13,11,15)(14,25,31,24)(17,27)(18,33)(19,29)(20,28)(22,35)(26,32)(34,36);(5,7,11,17,9,19,10,20)(6,12,15,27,16,29,13,28)(8,22,25,34,23,18,24,32)(14,26,21,35,31,36,30,33);(6,13,16,15)(7,17,19,20)(14,30,31,21)(18,32,22,34);(2,4)(5,8,10,24,9,23,11,25)(6,14,15,21,16,31,13,30)(7,18,20,34,19,22,17,32)(12,26,27,35,29,36,28,33)

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=fg,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=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, 353574);