#140675

order	512
nilpotency class	4
isoclinism	isoclinic class of #140675
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1964
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9652
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

378 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=e,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=di,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47871);