#145373

order	512
nilpotency class	4
isoclinism	isoclinic class of #145372
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 312 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=dh,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fgi,d^c=di,e^c=eghi,f^c=fi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52569);