#486325

order	512
nilpotency class	4
isoclinism	isoclinic class of #486320
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24279	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

868 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 868 conjugacy classes of subgroups — too many to draw. Only its 132 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 — 4 generators on 72 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=gi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393521);