#499319

order	512
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2067
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2067
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

741 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320511, #72320565, #72320745
maximal normal subgroup of	#72320511, #72320565, #72320745
maximal quotient of	#72322910

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406515);