#404880

order	512
nilpotency class	3
isoclinism	isoclinic class of #395172
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2431
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14905
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1086 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)(11,28)(14,41)(15,39)(17,40)(19,50)(20,34)(23,35)(26,36)(27,37)(29,43)(30,48)(31,49)(32,47)(33,52)(38,44)(42,60)(45,54)(46,56)(51,58)(53,59)(55,62)(57,64)(61,63);(1,3,6,16,7,17,28,49)(2,8,11,13,12,31,25,40)(4,14,26,47,21,44,20,43)(5,15,27,48,24,46,23,45)(9,32,36,41,18,29,34,38)(10,30,37,39,22,54,35,56)(19,42,51,62,52,63,53,64)(33,57,58,61,50,55,59,60);(1,4,6,26,7,21,28,20)(2,9,11,36,12,18,25,34)(3,14,16,47,17,44,49,43)(5,19,27,51,24,52,23,53)(8,29,13,38,31,32,40,41)(10,33,37,58,22,50,35,59)(15,42,48,62,46,63,45,64)(30,55,39,60,54,57,56,61);(1,5)(2,10,12,22)(3,15)(4,19)(6,27)(7,24)(8,30,31,54)(9,33,18,50)(11,37,25,35)(13,39,40,56)(14,42)(16,48)(17,46)(20,53)(21,52)(23,28)(26,51)(29,55,32,57)(34,59,36,58)(38,60,41,61)(43,64)(44,63)(45,49)(47,62)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285536, #72285720, #72285960, …
maximal normal subgroup of	#72285536, #72285720, #72285960, …
maximal quotient of	C₂×#404880

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312076);