#149684

order	512
nilpotency class	5
isoclinism	isoclinic class of #13569
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×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 ◁ 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:

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

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,c^b=ci,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56880);