#138215

order	512
nilpotency class	4
isoclinism	isoclinic class of #138213
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9142
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

369 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,13,7,11,30,31,8,14,15,45,5,12,27,42)(3,9,19,34,17,35,46,62,20,36,52,57,18,33,28,43)(4,21,24,53,23,41,47,44,26,40,29,56,25,10,37,39)(16,38,49,63,48,60,55,61,50,59,51,64,22,32,54,58);(1,3,15,46,7,18,27,19,8,20,6,28,5,17,30,52)(2,9,31,57,12,35,13,43,14,36,42,34,11,33,45,62)(4,22,29,49,23,50,37,54,26,48,24,51,25,16,47,55)(10,38,53,64,40,60,39,58,41,59,56,63,21,32,44,61);(1,4,5,25,8,26,7,23)(2,10,12,40,14,41,11,21)(3,16,18,22,20,50,17,48)(6,29,27,47,15,24,30,37)(9,32,33,59,36,60,35,38)(13,44,42,53,45,39,31,56)(19,51,28,55,52,49,46,54)(34,61,43,63,57,58,62,64)

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=h,d^2=gh,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45411);