#499040

order	512
nilpotency class	4
isoclinism	isoclinic class of #13402
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2034
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)(5,11)(6,25)(7,29)(8,13)(9,17)(10,22)(12,27)(15,42)(16,40)(18,48)(19,32)(21,53)(23,54)(24,38)(26,34)(28,57)(30,37)(31,59)(33,46)(35,55)(36,58)(39,45)(41,50)(43,62)(44,52)(47,56)(49,61)(51,63)(60,64);(1,3,7,19,8,17,5,16)(2,9,11,32,13,14,29,40)(4,15,21,43,23,44,30,51)(6,18,27,50,28,47,24,46)(10,31,36,61,35,39,34,60)(12,33,25,56,38,41,57,48)(20,52,37,62,54,42,53,63)(22,45,26,49,55,59,58,64);(1,4)(2,10)(3,15,17,44)(5,21)(6,26)(7,30)(8,23)(9,31,14,39)(11,34)(12,37)(13,35)(16,43,19,51)(18,49,47,64)(20,25)(22,27)(24,55)(28,58)(29,36)(32,60,40,61)(33,62,41,63)(38,53)(42,56,52,48)(45,46,59,50)(54,57);(1,5,8,7)(2,11,13,29)(3,16,17,19)(4,21,23,30)(6,24,28,27)(9,32,14,40)(10,34,35,36)(12,25,38,57)(15,43,44,51)(18,46,47,50)(20,53,54,37)(22,26,55,58)(31,60,39,61)(33,56,41,48)(42,62,52,63)(45,49,59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320498, #72320565, #72320752
maximal normal subgroup of	#72320498, #72320565, #72320752
maximal quotient of	#72322788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406236);