#149681

order	512
nilpotency class	5
isoclinism	isoclinic class of #1402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₈
Frattini, cosocle	#408	C₂³
socle	C₂	#11471
prime radical	#3018	C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

399 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=g,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,e^b=efi,f^b=fi,h^b=hi,e^d=ehi,f^d=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56877);