#480175

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,e^2=i,f^2=h,h^2=i,b^a=be,e^a=eg,e^b=ei,f^b=fgi,d^c=di

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, 387371);