#149655

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
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	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×C₂²⋊C₁₆
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

221 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 88 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)(10,15,16,13)(12,14)(17,18,20,25,21,26,33,43,22,27,34,44,36,46,55,65,23,28,35,45,37,47,56,66,38,48,57,67,58,68,73,78)(19,24,30,40,31,41,52,62,32,42,53,63,54,64,72,77,29,39,49,59,50,60,69,74,51,61,70,75,71,76,79,80);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,30)(21,31)(22,32)(23,29)(25,40)(26,41)(27,42)(28,39)(33,52)(34,53)(35,49)(36,54)(37,50)(38,51)(43,62)(44,63)(45,59)(46,64)(47,60)(48,61)(55,72)(56,69)(57,70)(58,71)(65,77)(66,74)(67,75)(68,76)(73,79)(78,80);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,29)(24,39)(30,49)(31,50)(32,51)(40,59)(41,60)(42,61)(52,69)(53,70)(54,71)(62,74)(63,75)(64,76)(72,79)(77,80)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56851);