#152923

order	512
nilpotency class	5
isoclinism	isoclinic class of #152923
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13221
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#24287	C₂
LCS	G ▷ C₂×#41 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 445 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)(3,14,19,38,20,41,18,50)(4,21)(5,26,15,39)(6,29)(7,12)(8,13)(9,17,33,27,35,49,34,42)(10,23)(11,28,31,30)(16,44,47,62,48,63,46,40)(22,36,43,58)(24,37)(25,53)(32,59,51,45,61,54,60,64)(52,55,57,56);(1,3,15,42)(2,9,31,50)(4,16,43,59)(5,27,8,20)(6,18,30,49)(7,19,28,17)(10,32,58,63)(11,38,13,35)(12,33,26,14)(21,51,57,40)(22,54,25,48)(23,46,56,64)(24,47,55,45)(29,34,39,41)(36,44,37,61)(52,62,53,60);(1,4,6,23,8,25,7,24)(2,10,12,21,13,37,29,53)(3,16,18,46,20,48,19,47)(5,22,28,55,15,43,30,56)(9,32,33,51,35,61,34,60)(11,36,39,52,31,58,26,57)(14,40,38,44,41,62,50,63)(17,45,42,59,49,64,27,54)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=e,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,d^b=dg,f^b=fg,g^b=gh,h^b=hi,f^d=fhi,g^d=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, 60119);