#6470425

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei

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, 6377621);