#462119

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
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

573 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)(5,6,12,16,11,17,10,28)(7,18,27,30,20,15,9,13)(8,21,23,32,25,31,24,14)(19,35,36,33,22,29,26,34);(5,7,10,9,11,20,12,27)(6,13,16,18,17,30,28,15)(8,22,24,36,25,19,23,26)(14,29,21,34,32,35,31,33);(5,8,10,24,11,25,12,23)(6,14,16,21,17,32,28,31)(7,19,9,26,20,22,27,36)(13,29,18,34,30,35,15,33);(2,4)(5,9,10,20,11,27,12,7)(6,15,16,13,17,18,28,30)(8,22,23,26,25,19,24,36)(14,29,31,33,32,35,21,34)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369315);