#464307

order	512
nilpotency class	3
isoclinism	isoclinic class of #455155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1353 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^d=hi

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, 371503);