#140788

order	512
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9665
prime radical	#3414	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

313 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cgh,d^b=di,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320349, #72320718, #72320780
maximal normal subgroup of	#72320349, #72320718, #72320780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47984);