#489389

order	512
nilpotency class	4
isoclinism	isoclinic class of #489385
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	#62437	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 749 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396585);