#449136

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

592 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)(5,6,11,18,12,17,10,16)(7,19,22,32,24,31,23,13)(8,25,9,28,27,15,26,14)(20,30,34,33,21,35,29,36);(5,7,11,23,12,24,10,22)(6,13,16,19,17,32,18,31)(8,20,26,34,27,21,9,29)(14,30,25,36,28,35,15,33);(5,8,10,26,12,27,11,9)(6,14,16,15,17,28,18,25)(7,20,22,34,24,21,23,29)(13,30,19,33,32,35,31,36);(2,4)(5,9,11,27,12,26,10,8)(6,15,18,14,17,25,16,28)(7,21,22,29,24,20,23,34)(13,30,19,33,32,35,31,36)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=e,d^2=eg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^c=hi,f^d=fi,h^d=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, 356332);