#138554

order	512
nilpotency class	4
isoclinism	isoclinic class of #138552
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#121	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

292 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

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

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, 45750);