#499537

order	512
nilpotency class	4
isoclinism	isoclinic class of #498863
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#493
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2093
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

761 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320511, #72320528, #72320541, …
maximal normal subgroup of	#72320511, #72320528, #72320541, …
maximal quotient of	#72322897, #72322957

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406733);