#498040

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=deh,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405236);