#475048

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1013 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)(3,6)(4,5)(7,8)(9,10,14,20,16,22,15,21)(11,23,27,17,28,41,26,42)(12,29,32,18,34,45,33,46)(13,19,37,49,38,50,36,48)(24,39,30,43,54,59,53,66)(25,51,56,69,58,68,57,40)(31,47,62,44,63,60,35,64)(52,71,55,70,72,65,61,67);(2,5)(6,8)(9,11)(10,17,22,42)(12,30,34,53)(13,25)(14,26)(15,27)(16,28)(18,43)(19,40,50,69)(20,23,21,41)(24,33,54,32)(29,59)(31,61,63,55)(35,52,62,72)(36,56)(37,57)(38,58)(39,45)(44,70)(46,66)(47,65)(48,51,49,68)(60,71)(64,67);(1,3)(2,6)(4,7)(5,8)(9,12)(10,18)(11,24)(13,35)(14,33)(15,32)(16,34)(17,39)(19,47)(20,29)(21,45)(22,46)(23,43)(25,55)(26,30)(27,53)(28,54)(31,36)(37,63)(38,62)(40,67)(41,66)(42,59)(44,48)(49,64)(50,60)(51,65)(52,56)(57,72)(58,61)(68,71)(69,70);(3,7)(6,8)(9,13,15,37,16,38,14,36)(10,19,21,49,22,50,20,48)(11,25,27,57,28,58,26,56)(12,31,33,62,34,63,32,35)(17,40,41,68,42,69,23,51)(18,44,29,60,46,64,45,47)(24,52,30,61,54,72,53,55)(39,65,43,70,59,71,66,67)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,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, 382244);