#403462

order	512
nilpotency class	3
isoclinism	isoclinic class of #395172
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2431
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14875
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285582, #72285720, #72285960, …
maximal normal subgroup of	#72285582, #72285720, #72285960, …
maximal quotient of	C₂×#403462

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310658);