#149683

order	512
nilpotency class	5
isoclinism	isoclinic class of #1404
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×Q₈	C₂²×C₈
Frattini, cosocle	#408	C₂³
socle	C₂	#11471
prime radical	#3018	C₄
LCS	G ▷ C₂×Q₈ ▷ 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=g,d^2=i,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=di,e^b=ef,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, 56879);