#403463

order	512
nilpotency class	3
isoclinism	isoclinic class of #395191
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

954 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)(3,15)(4,21)(5,25)(6,14)(7,12)(8,13)(9,18)(10,24)(11,28)(16,40)(17,31)(19,33)(20,32)(22,50)(23,36)(26,38)(27,37)(29,39)(30,43)(34,48)(35,54)(41,46)(42,58)(44,57)(45,53)(47,56)(49,51)(52,59)(55,62)(60,61)(63,64);(1,3)(2,9,13,32)(4,16)(5,17)(6,18)(7,20)(8,19)(10,34,21,30)(11,31,38,56)(12,15,14,33)(22,42)(23,43)(24,44)(25,53,37,41)(26,45)(27,47)(28,46)(29,48)(35,58,49,55)(36,57,39,40)(50,60,59,64)(51,61)(52,62)(54,63);(1,4)(2,10,13,21)(3,16)(5,22)(6,23)(7,29)(8,24)(9,30,32,34)(11,35,38,49)(12,36,14,39)(15,40,33,57)(17,42)(18,43)(19,44)(20,48)(25,50,37,59)(26,51)(27,52)(28,54)(31,55,56,58)(41,60,53,64)(45,61)(46,63)(47,62);(1,5,6,26,8,27,7,28)(2,11,12,37,13,38,14,25)(3,17,18,45,19,47,20,46)(4,22,23,51,24,52,29,54)(9,31,15,41,32,56,33,53)(10,35,36,59,21,49,39,50)(16,42,43,61,44,62,48,63)(30,55,40,60,34,58,57,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285525, #72285720, #72285996, …
maximal normal subgroup of	#72285525, #72285720, #72285996, …
maximal quotient of	C₂×#403463

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310659);