#146221

order	512
nilpotency class	4
isoclinism	isoclinic class of #146221
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=de,d^2=gi,e^2=g,f^2=hi,g^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=egi,f^b=fh,g^b=gi,d^c=dgi,e^c=egi,f^c=fhi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53417);