#511542

order	512
nilpotency class	4
isoclinism	isoclinic class of #511541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32659
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

403 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efh,d^2=efi,f^2=i,g^2=h,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=ghi,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	#72320511, #72320520, #72320582
maximal normal subgroup of	#72320511, #72320520, #72320582
maximal quotient of	#72321294, #72321477, #72321658, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418738);