#488759

order	512
nilpotency class	4
isoclinism	isoclinic class of #488723
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

759 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 759 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395955);