#493820

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×#1808
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

941 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)(4,19,21,9)(5,23)(6,26)(7,11)(8,17)(10,14)(12,24)(15,40,41,33)(16,34)(18,46)(20,50,52,53)(22,31,51,49)(25,54)(27,42,47,62)(28,44)(29,56)(30,38,48,60)(32,39)(35,37)(36,45)(43,58,63,61)(55,64,57,59);(1,3,14,37,6,18,25,44,7,16,39,56,24,45,5,17)(2,8,23,36,12,29,32,34,11,28,54,46,26,35,10,13)(4,15,38,59,22,43,52,62,21,41,60,64,51,63,20,42)(9,27,53,58,31,55,48,40,19,47,50,61,49,57,30,33);(1,4)(2,9)(3,15)(5,20)(6,22)(7,21)(8,27)(10,30)(11,19)(12,31)(13,33)(14,38)(16,41)(17,42)(18,43)(23,53)(24,51)(25,52)(26,49)(28,47)(29,55)(32,48)(34,40)(35,57)(36,58)(37,59)(39,60)(44,62)(45,63)(46,61)(50,54)(56,64);(3,16)(8,28)(13,34)(15,41)(17,44)(18,45)(27,47)(29,35)(33,40)(36,46)(37,56)(42,62)(43,63)(55,57)(58,61)(59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307189, #72307247, #72307277, …
maximal normal subgroup of	#72307189, #72307247, #72307277, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401016);