#137228

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1874
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

415 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(17,18,23,30,24,31,48,62,25,32,47,61,22,29,21,28)(19,33,37,55,38,59,35,52,39,66,45,26,36,53,73,54)(20,40,43,60,42,58,46,64,44,63,50,56,49,27,41,57)(34,67,70,78,69,77,68,79,71,80,75,76,74,65,72,51);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,19,22,38,25,39,24,36)(18,26,29,54,32,55,31,52)(20,34,49,69,44,71,42,74)(21,45,47,73,48,37,23,35)(27,51,63,78,58,79,40,76)(28,59,61,66,62,53,30,33)(41,75,50,72,46,70,43,68)(56,80,64,65,60,67,57,77);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,20)(18,27)(19,34)(21,46)(22,49)(23,50)(24,42)(25,44)(26,51)(28,60)(29,63)(30,64)(31,40)(32,58)(33,65)(35,72)(36,74)(37,75)(38,69)(39,71)(41,48)(43,47)(45,70)(52,76)(53,80)(54,78)(55,79)(56,62)(57,61)(59,67)(66,77)(68,73)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=gi,e^2=i,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=ehi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44424);