#488750

order	512
nilpotency class	4
isoclinism	isoclinic class of #488707
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	#62341	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

763 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 763 conjugacy classes of subgroups — too many to draw. Only its 134 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)(11,25,31,57)(12,32,36,55)(13,21,38,51)(14,20,37,52)(15,23,16,22)(18,42,47,28)(19,41,50,35)(26,58,62,69)(27,46,39,45)(29,53,30,44)(33,48,67,66)(34,54,40,49)(43,59,72,68)(56,64,70,61)(60,65,63,71);(1,2)(3,8)(4,6)(5,7)(9,11,17,31)(10,18,24,47)(12,26,36,62)(13,28,38,42)(14,39,37,27)(15,30,16,29)(19,43,50,72)(20,25,52,57)(21,53,51,44)(22,46,23,45)(32,65,55,71)(33,63,67,60)(34,59,40,68)(35,64,41,61)(48,56,66,70)(49,69,54,58);(3,7)(5,8)(9,12)(10,19)(11,26)(13,33)(14,40)(15,41)(16,35)(17,36)(18,43)(20,48)(21,54)(22,55)(23,32)(24,50)(25,56)(27,59)(28,63)(29,64)(30,61)(31,62)(34,37)(38,67)(39,68)(42,60)(44,69)(45,65)(46,71)(47,72)(49,51)(52,66)(53,58)(57,70);(2,6)(5,8)(9,13,16,37,17,38,15,14)(10,20,23,51,24,52,22,21)(11,27,30,28,31,39,29,42)(12,33,35,34,36,67,41,40)(18,44,46,25,47,53,45,57)(19,48,32,49,50,66,55,54)(26,59,61,63,62,68,64,60)(43,69,71,56,72,58,65,70)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=ghi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgh,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395946);