#499027

order	512
nilpotency class	4
isoclinism	isoclinic class of #13391
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2033
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1043 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)(3,14)(4,20)(5,10)(6,28)(7,13)(8,17)(9,23)(11,27)(12,26)(15,44)(16,34)(18,32)(19,33)(21,55)(22,53)(24,40)(25,54)(29,36)(30,39)(31,56)(35,63)(37,57)(38,58)(41,59)(42,46)(43,48)(45,64)(47,52)(49,60)(50,61)(51,62);(1,3)(2,8)(4,21)(5,16)(6,18)(7,19)(9,35)(10,14)(11,33)(12,32)(13,34)(15,22)(17,27)(20,52)(23,56)(24,47)(25,45)(26,48)(28,60)(29,46)(30,51)(31,36)(37,61)(38,42)(39,43)(40,64)(41,62)(44,54)(49,59)(50,58)(53,55)(57,63);(1,4,5,22,7,25,27,24)(2,9,11,37,13,38,10,36)(3,15,17,21,19,47,16,45)(6,29,26,58,30,57,59,23)(8,31,14,42,34,61,33,35)(12,40,28,54,41,53,39,20)(18,50,49,46,51,56,48,63)(32,52,43,55,62,44,60,64);(1,5,7,27)(2,10,13,11)(3,16,19,17)(4,22,25,24)(6,26,30,59)(8,14,34,33)(9,36,38,37)(12,39,41,28)(15,45,47,21)(18,48,51,49)(20,53,54,40)(23,29,58,57)(31,42,61,35)(32,43,62,60)(44,64,52,55)(46,50,63,56)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320541, #72320565, #72320763
maximal normal subgroup of	#72320541, #72320565, #72320763
maximal quotient of	#72322989

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406223);