#150792

order	512
nilpotency class	5
isoclinism	isoclinic class of #150785
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

209 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 74 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)(17,18,21,26,23,29,42,28)(19,30,32,60,35,61,34,69)(20,36,39,70,41,59,75,71)(22,43,38,27,45,55,67,57)(24,46,47,64,49,63,58,31)(25,50,52,62,54,76,53,40)(33,66,65,78,68,79,44,51)(37,56,72,77,74,80,73,48);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19,23,35)(18,24,29,49)(20,37,41,74)(21,32,42,34)(22,44,45,65)(25,51,54,78)(26,47,28,58)(27,56,57,80)(30,59,61,36)(31,62,64,40)(33,67,68,38)(39,72,75,73)(43,48,55,77)(46,76,63,50)(52,66,53,79)(60,71,69,70);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,38,50,21,39,45,62,23,41,67,76,42,75,22,40)(18,25,43,71,28,53,57,59,29,54,55,70,26,52,27,36)(19,31,44,73,34,63,68,74,35,64,65,72,32,46,33,37)(24,30,56,66,47,60,77,78,49,61,80,79,58,69,48,51)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57988);