#149657

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

214 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 86 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)(10,15,16,13)(12,14)(17,18,21,28,22,29,40,55,23,30,41,56,43,57,68,77,24,26,42,49,44,50,69,73,20,27,37,52,38,53,65,75)(19,31,33,58,34,59,62,78,32,46,60,54,61,71,79,76,35,51,39,66,63,74,67,80,36,25,45,47,64,48,70,72);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,25)(20,36)(21,39)(22,34)(23,32)(24,35)(26,46)(27,51)(28,54)(29,48)(30,31)(33,42)(37,60)(38,64)(40,67)(41,45)(43,61)(44,63)(47,49)(50,71)(52,58)(53,74)(55,76)(56,66)(57,59)(62,69)(65,79)(68,70)(72,73)(75,78)(77,80);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,26)(25,46)(27,30)(28,49)(29,50)(31,51)(47,54)(48,71)(52,56)(53,57)(55,73)(58,66)(59,74)(72,76)(75,77)(78,80)

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=df,d^b=di,e^b=efi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56853);