#137787

order	512
nilpotency class	4
isoclinism	isoclinic class of #137786
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1899
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9084
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 380 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dgi,e^b=ei,f^b=fgi,g^b=gi,d^c=di,e^c=ei,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, 44983);