#152953

order	512
nilpotency class	5
isoclinism	isoclinic class of #152953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13224
commutator	C₂×C₄²	C₂²×C₄
Frattini, prime radical, cosocle	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

297 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,15,19,40)(4,21)(5,12)(6,28)(7,13,31,41)(8,14)(9,17,34,18)(10,23)(11,27,32,29)(16,45,48,59)(20,44,30,43)(22,39)(24,51,57,37)(25,52)(26,36,50,35)(33,56,58,49)(38,54,61,55)(42,46,60,47)(53,63,64,62);(1,3,6,18,8,20,5,26)(2,9,32,43,14,36,11,40)(4,16,23,47,25,49,22,53)(7,30,27,17,31,19,29,50)(10,33,51,62,39,45,37,60)(12,15,41,34,28,44,13,35)(21,42,61,58,52,63,38,59)(24,56,54,46,57,48,55,64);(1,4,6,23,8,25,5,22)(2,10,12,21,14,39,28,52)(3,16,18,47,20,49,26,53)(7,24,29,55,31,57,27,54)(9,33,15,42,36,45,44,63)(11,37,13,38,32,51,41,61)(17,46,30,56,50,64,19,48)(34,58,40,60,35,59,43,62)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=e,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dgh,f^b=fghi,g^b=ghi,h^b=hi,f^d=fhi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60149);