#150796

order	512
nilpotency class	5
isoclinism	isoclinic class of #150788
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

207 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2)(3,9,11,6)(4,7)(5,8)(10,15,12,14)(13,16)(17,18,21,28,24,32,48,31)(19,33,35,69,38,54,46,70)(20,39,42,76,45,63,80,77)(22,49,47,29,23,51,41,30)(25,53,55,34,59,71,58,73)(26,60,62,43,27,65,64,44)(36,74,50,61,37,75,52,66)(40,67,78,56,72,68,79,57);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19,24,38)(18,25,32,59)(20,40,45,72)(21,46,48,35)(22,50,23,52)(26,61,27,66)(28,58,31,55)(29,67,30,68)(33,39,54,63)(34,43,73,44)(36,41,37,47)(42,79,80,78)(49,56,51,57)(53,65,71,60)(62,74,64,75)(69,77,70,76);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,41,60,21,42,22,43,24,45,47,65,48,80,23,44)(18,26,51,77,31,64,29,63,32,27,49,76,28,62,30,39)(19,34,52,78,35,71,36,72,38,73,50,79,46,53,37,40)(25,54,68,74,58,69,56,66,59,33,67,75,55,70,57,61)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57992);