#6432673

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1693 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)(3,9)(4,19,22,38)(5,11)(6,12)(7,28)(8,13)(10,23,35,24)(14,33,40,56)(15,30)(16,31)(17,32)(18,45)(20,29,41,46)(21,47,49,59)(25,36)(26,52)(27,37)(34,50,58,51)(39,57,60,64)(42,54)(43,55)(44,62)(48,53,61,63);(1,3)(2,9)(4,20,22,41)(5,15)(6,17)(7,18)(8,16)(10,33,35,56)(11,30)(12,32)(13,31)(14,24,40,23)(19,46,38,29)(21,48,49,61)(25,43)(26,44)(27,42)(28,45)(34,57,58,64)(36,55)(37,54)(39,51,60,50)(47,63,59,53)(52,62);(1,4)(2,10)(3,14)(5,21)(6,24)(7,23)(8,22)(9,29)(11,34)(12,38)(13,35)(15,39)(16,40)(17,20)(18,41)(19,28)(25,51)(26,50)(27,49)(30,53)(31,46)(32,56)(33,45)(36,59)(37,58)(42,60)(43,48)(44,61)(47,52)(54,63)(55,64)(57,62);(1,5,6,25,8,27,7,26)(2,11,12,36,13,37,28,52)(3,15,17,43,16,42,18,44)(4,21,24,51,22,49,23,50)(9,30,32,55,31,54,45,62)(10,34,38,59,35,58,19,47)(14,39,20,48,40,60,41,61)(29,53,56,64,46,63,33,57);(3,16)(9,31)(14,40)(15,42)(17,18)(20,41)(29,46)(30,54)(32,45)(33,56)(39,60)(43,44)(48,61)(53,63)(55,62)(57,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249829, #72285525, #72285642, …
maximal normal subgroup of	#72249829, #72285525, #72285642, …
maximal quotient of	#72257150, C₂×#6432673

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339869);