#497975

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 ◁ #1752 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

570 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,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, 405171);