#136606

order	512
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#102
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(6,9,11,12)(13,14,18,24,19,25,34,41,20,26,32,39,16,22,17,23)(15,27,30,38,28,40,29,42,31,43,33,21,35,37,44,36);(5,6,10,9,8,11,7,12)(13,15,16,28,20,31,19,35)(14,21,22,36,26,38,25,42)(17,33,32,44,34,30,18,29)(23,40,39,43,41,37,24,27);(2,4)(13,16,20,19)(14,22,26,25)(15,28,31,35)(17,32,34,18)(21,36,38,42)(23,39,41,24)(27,40,43,37)(29,33,44,30)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^b=dg,f^b=fg,g^b=gi,h^b=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43802);