#475045

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

1017 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)(11,23,28,41)(12,29)(13,19,38,50)(14,20)(15,21)(16,22)(17,26,42,27)(18,33)(24,43,54,59)(25,51)(30,61,53,39)(31,44,64,60)(32,46)(34,45)(35,47,63,70)(36,48,37,49)(40,58)(52,66)(55,72)(56,67)(57,68)(62,65)(69,71);(2,5)(6,8)(9,11)(10,17,21,42)(12,30,33,53)(13,25)(14,26)(15,28)(16,27)(18,43)(19,40,50,67)(20,23,22,41)(24,34,54,32)(29,59)(31,62,64,55)(35,52,63,71)(36,56)(37,58)(38,57)(39,45)(44,69)(46,61)(47,65)(48,51,49,68)(60,66)(70,72);(1,3)(2,6)(4,7)(5,8)(9,12)(10,18)(11,24)(13,35)(14,34)(15,33)(16,32)(17,39)(19,47)(20,46)(21,29)(22,45)(23,43)(25,55)(26,30)(27,53)(28,54)(31,36)(37,64)(38,63)(40,66)(41,59)(42,61)(44,48)(49,60)(50,70)(51,65)(52,56)(57,62)(58,71)(67,69)(68,72);(3,7)(6,8)(9,13,16,37,15,38,14,36)(10,19,22,49,21,50,20,48)(11,25,27,58,28,57,26,56)(12,31,34,63,33,64,32,35)(17,40,41,68,42,67,23,51)(18,44,46,70,29,60,45,47)(24,52,30,62,54,71,53,55)(39,65,43,69,61,72,59,66)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=g,e^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,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, 382241);