#493805

order	512
nilpotency class	4
isoclinism	isoclinic class of #483412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#427
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1807
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

933 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)(3,13,15,34)(4,18,20,9)(5,22)(6,25)(7,11)(8,16,28,40)(10,23)(12,26)(14,36)(17,43,41,35)(19,47,48,51)(21,31,50,46)(24,52)(27,38)(29,56,54,42)(30,49,45,63)(32,53)(33,37)(39,57)(44,60)(55,64)(58,62)(59,61);(1,3,5,16,6,17,23,42,7,15,24,40,26,41,53,56)(2,8,10,13,12,29,22,35,11,28,32,34,25,54,52,43)(4,14,19,38,21,39,49,62,20,37,48,60,50,61,63,64)(9,27,30,33,31,55,51,57,18,44,45,36,46,58,47,59);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,27)(10,30)(11,18)(12,31)(13,33)(15,37)(16,38)(17,39)(22,51)(23,49)(24,48)(25,46)(26,50)(28,44)(29,55)(32,45)(34,36)(35,57)(40,60)(41,61)(42,62)(43,59)(47,52)(53,63)(54,58)(56,64);(3,15)(8,28)(13,34)(14,37)(16,40)(17,41)(27,44)(29,54)(33,36)(35,43)(38,60)(39,61)(42,56)(55,58)(57,59)(62,64)

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=di,g^b=gi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307235, #72307247, #72307292, …
maximal normal subgroup of	#72307235, #72307247, #72307292, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401001);