#150787

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	#2215	C₂²
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

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

normal subgroups only

This group has 219 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) = 128

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

Polycyclic presentation:

2^9|a^2=e,c^2=fhi,d^2=gh,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, 57983);