#6461517

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442597
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2598
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1823 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)(7,12)(8,13)(9,17)(15,42)(16,38)(18,41)(19,32)(20,30)(21,34)(22,36)(23,37)(24,52)(25,39)(26,53)(27,40)(29,35)(31,46)(33,44)(43,63)(45,62)(47,56)(48,55)(49,57)(50,59)(51,61)(54,60)(58,64);(1,3)(2,9)(4,20)(5,24)(6,19)(7,17)(8,18)(10,33)(11,38)(12,41)(13,32)(14,28)(15,22)(16,27)(21,49)(23,47)(25,46)(26,45)(29,44)(30,36)(31,40)(34,58)(35,56)(37,42)(39,62)(43,50)(48,51)(52,53)(54,64)(55,59)(57,61)(60,63);(1,4,7,22,8,23,6,29)(2,10,12,36,13,37,28,35)(3,15,19,20,18,44,17,47)(5,21,27,51,25,50,26,54)(9,30,14,33,32,56,41,42)(11,34,40,61,39,59,53,60)(16,43,24,48,45,49,46,64)(31,55,38,57,52,58,62,63);(1,5)(2,11)(3,16,18,45)(4,21)(6,26)(7,27)(8,25)(9,31,32,52)(10,34)(12,40)(13,39)(14,38,41,62)(15,43,44,49)(17,46,19,24)(20,48,47,64)(22,51)(23,50)(28,53)(29,54)(30,55,56,58)(33,57,42,63)(35,60)(36,61)(37,59);(5,25)(11,39)(16,45)(21,50)(24,46)(26,27)(31,52)(34,59)(38,62)(40,53)(43,49)(48,64)(51,54)(55,58)(57,63)(60,61)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285536, #72285642, #72285996, …
maximal normal subgroup of	#72285536, #72285642, #72285996, …
maximal quotient of	C₂×#6461517

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368713);