#151001

order	512
nilpotency class	5
isoclinism	isoclinic class of #150788
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂³	#357
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

187 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 187 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=gh,e^2=fghi,g^2=hi,h^2=i,b^a=bd,c^a=cfi,d^a=df,d^b=dg,e^b=efg,g^b=gh,h^b=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58197);