#400053

order	512
nilpotency class	3
isoclinism	isoclinic class of #395190
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2431
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14795
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1360 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)(4,9)(5,21)(6,24)(7,12)(8,16)(10,23)(11,27)(14,40)(15,39)(17,36)(18,32)(19,49)(20,34)(22,35)(25,30)(26,37)(28,50)(29,46)(31,42)(33,52)(38,43)(41,60)(44,53)(45,56)(47,59)(48,54)(51,58)(55,64)(57,62)(61,63);(1,3)(2,8)(4,14)(5,15)(6,25)(7,17)(9,31)(10,29)(11,36)(12,30)(13,24)(16,27)(18,41)(19,50)(20,43)(21,53)(22,46)(23,45)(26,44)(28,34)(32,57)(33,38)(35,39)(37,56)(40,49)(42,52)(47,64)(48,63)(51,62)(54,55)(58,61)(59,60);(1,4)(2,9)(3,14)(5,18)(6,19)(7,20)(8,28)(10,32)(11,33)(12,34)(13,38)(15,41)(16,42)(17,43)(21,54)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(29,55)(30,31)(35,58)(36,40)(37,59)(39,60)(44,62)(45,63)(46,64)(53,57)(56,61);(1,5,6,26,7,23,27,22)(2,10,11,37,12,21,24,35)(3,15,25,44,17,45,16,46)(4,18,19,51,20,48,52,47)(8,29,36,56,30,53,13,39)(9,32,33,59,34,54,49,58)(14,41,50,62,43,63,42,64)(28,55,40,61,31,57,38,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307249);