#150964

order	512
nilpotency class	5
isoclinism	isoclinic class of #7453
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1365
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂	C₂×#1365
prime radical	C₂³	#361
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

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

normal subgroups only

This group has 193 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58160);