#149682

order	512
nilpotency class	5
isoclinism	isoclinic class of #13561
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×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 ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 397 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:

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

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,c^b=ci,e^b=efi,f^b=fi,h^b=hi,e^d=ehi,f^d=fi,f^e=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, 56878);