#150998

order	512
nilpotency class	5
isoclinism	isoclinic class of #150785
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

190 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 76 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)(17,18,21,27,43,59,40,56,23,29,41,57,22,28,44,60,24,30,45,61,38,54,73,79,47,63,20,26,39,55,74,80)(19,31,33,50,69,58,72,52,35,65,32,62,34,53,42,64,36,66,67,77,46,75,37,25,48,49,68,76,70,78,71,51);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,25)(20,37)(21,42)(22,46)(23,48)(24,36)(26,53)(27,58)(28,62)(29,64)(30,52)(31,56)(32,40)(33,44)(34,38)(35,47)(39,69)(41,72)(43,70)(45,71)(49,60)(50,54)(51,63)(55,76)(57,78)(59,77)(61,75)(65,80)(66,79)(67,74)(68,73);(2,7)(6,9)(8,15)(14,16)

Polycyclic presentation:

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

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, 58194);