#147389

order	512
nilpotency class	3
isoclinism	isoclinic class of #105681
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1752
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^c=ei,f^c=fi,g^c=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, 54585);