#511428

order	512
nilpotency class	4
isoclinism	isoclinic class of #511427
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32554
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2961 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

395 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,d^2=ef,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320520, #72320582, #72320615
maximal normal subgroup of	#72320520, #72320582, #72320615
maximal quotient of	#72321421, #72321592, #72321668, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418624);