#464712

order	512
nilpotency class	3
isoclinism	isoclinic class of #454399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1448 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:

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

Polycyclic presentation:

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

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, 371908);