#6462905

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442595
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2535
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249829, #72249832, #72249836, …
maximal normal subgroup of	#72249829, #72249832, #72249836, …
maximal quotient of	#72257124, #72322857, #72322924

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370101);