#150950

order	512
nilpotency class	5
isoclinism	isoclinic class of #7447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1361
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂	C₂×#1361
prime radical	#33713	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

279 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=ghi,e^2=fghi,f^2=hi,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,h^a=hi,d^b=dg,e^b=efgh,f^b=fh,g^b=ghi,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225338, #72225339, #72225343, …
maximal normal subgroup of	#72225338, #72225339, #72225343, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58146);