#146259

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:

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

Polycyclic presentation:

2^9|a^2=dh,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, 53455);