#93718

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂²×C₄	C₂×C₁₆
Frattini, cosocle	C₂×C₄⋊C₁₆	C₂²
socle	C₂²	#1322
prime radical	C₂³	C₂²⋊C₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂×C₄⋊C₁₆ ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

109 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=gi,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,c^b=ci,d^b=degi,d^c=dg

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, 914);