#498044

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
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

528 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:

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

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,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405240);