#489187

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

761 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 761 conjugacy classes of subgroups — too many to draw. Only its 148 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)(9,10,16,22)(11,23,27,51)(12,18,30,44)(13,21,15,19)(14,33,36,46)(17,38,41,26)(20,32,48,35)(24,50,55,71)(25,56,37,47)(28,45,31,42)(29,58,60,69)(34,49,62,40)(39,63,66,53)(43,61,68,57)(52,67,54,70)(59,64,72,65);(3,7)(5,8)(9,11,16,27)(10,17,22,41)(12,24,30,55)(13,25,15,37)(14,34,36,62)(18,39,44,66)(19,40,21,49)(20,47,48,56)(23,46,51,33)(26,32,38,35)(28,52,31,54)(29,59,60,72)(42,64,45,65)(43,67,68,70)(50,69,71,58)(53,61,63,57);(1,2,4,6)(3,8,7,5)(9,12)(10,18)(11,24)(13,31)(14,29)(15,28)(16,30)(17,39)(19,45)(20,43)(21,42)(22,44)(23,50)(25,54)(26,53)(27,55)(32,61)(33,58)(34,59)(35,57)(36,60)(37,52)(38,63)(40,65)(41,66)(46,69)(47,67)(48,68)(49,64)(51,71)(56,70)(62,72);(9,13,16,15)(10,19,22,21)(11,25,27,37)(12,28,30,31)(14,32,36,35)(17,40,41,49)(18,42,44,45)(20,46,48,33)(23,47,51,56)(24,52,55,54)(26,34,38,62)(29,57,60,61)(39,64,66,65)(43,58,68,69)(50,70,71,67)(53,72,63,59)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396383);