#486323

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	#62274	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 872 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)(9,10,17,24)(11,25,30,55)(12,31,35,52)(13,21,37,49)(14,20,36,50)(15,23,16,22)(18,41,45,28)(19,39,48,34)(26,56,61,71)(27,44,38,53)(29,43,40,51)(32,46,65,70)(33,63,64,47)(42,66,69,57)(54,60,72,59)(58,62,67,68);(1,2)(3,5)(4,6)(7,8)(9,11)(10,18)(12,26)(13,28)(14,38)(15,29)(16,40)(17,30)(19,42)(20,25)(21,51)(22,44)(23,53)(24,45)(27,36)(31,62)(32,58)(33,66)(34,59)(35,61)(37,41)(39,60)(43,49)(46,54)(47,71)(48,69)(50,55)(52,68)(56,63)(57,64)(65,67)(70,72);(1,3)(2,5)(4,7)(6,8)(9,12,17,35)(10,19,24,48)(11,26,30,61)(13,32,37,65)(14,33,36,64)(15,39,16,34)(18,42,45,69)(20,46,50,70)(21,47,49,63)(22,52,23,31)(25,54,55,72)(27,57,38,66)(28,58,41,67)(29,60,40,59)(43,56,51,71)(44,68,53,62);(9,13,16,36,17,37,15,14)(10,20,23,49,24,50,22,21)(11,27,29,28,30,38,40,41)(12,32,34,64,35,65,39,33)(18,43,44,25,45,51,53,55)(19,46,31,63,48,70,52,47)(26,57,60,58,61,66,59,67)(42,56,68,54,69,71,62,72)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=g,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, 393519);