#150978

order	512
nilpotency class	5
isoclinism	isoclinic class of #150977
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11952
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂³	#361
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

188 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fh,d^2=gh,e^2=fgh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58174);