#7622991

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#466
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂	C₂²×#466
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1897 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)(3,15)(4,22)(5,11)(6,28)(7,13)(8,14)(9,20)(10,25)(12,16)(17,45)(18,40)(19,41)(21,42)(23,35)(24,55)(26,38)(27,37)(29,36)(30,57)(31,50)(32,51)(33,43)(34,56)(39,49)(44,53)(46,62)(47,61)(48,52)(54,60)(58,64)(59,63);(1,3,16,43,7,21,29,50,8,18,27,51,5,19,6,20)(2,9,28,41,11,32,37,40,14,31,36,42,13,33,12,15)(4,17,44,57,25,49,56,63,26,46,54,64,23,47,24,48)(10,30,53,45,22,52,55,61,35,58,60,62,38,59,34,39);(1,4)(2,10)(3,17)(5,23)(6,24)(7,25)(8,26)(9,30)(11,22)(12,34)(13,38)(14,35)(15,39)(16,44)(18,46)(19,47)(20,48)(21,49)(27,54)(28,53)(29,56)(31,58)(32,52)(33,59)(36,60)(37,55)(40,61)(41,45)(42,62)(43,57)(50,63)(51,64);(3,18)(9,31)(15,40)(17,46)(19,21)(20,50)(30,58)(32,33)(39,61)(41,42)(43,51)(45,62)(47,49)(48,63)(52,59)(57,64);(1,5,8,7)(2,11,14,13)(3,19,18,21)(4,23,26,25)(6,27,29,16)(9,32,31,33)(10,22,35,38)(12,28,37,36)(15,41,40,42)(17,47,46,49)(20,51,50,43)(24,54,56,44)(30,52,58,59)(34,53,55,60)(39,45,61,62)(48,64,63,57)

Polycyclic presentation:

2^9|b^2=fh,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,h^a=hi,d^b=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285648, #72285705, #72285749, …
maximal normal subgroup of	#72285648, #72285705, #72285749, …
maximal quotient of	#72321876, #72323044, #72342546

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530187);