#386038

order	512
nilpotency class	3
isoclinism	isoclinic class of #386038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14419
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60645	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1563 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)(3,14,18,41)(4,20)(5,26,29,39)(6,12)(7,31)(8,13)(9,17,34,19)(10,24)(11,30,40,28)(15,35,44,48)(16,45)(21,32,43,59)(22,52,55,62)(23,50)(25,37)(27,56)(33,47)(36,54,49,53)(38,46)(42,58)(51,64)(57,60)(61,63);(1,3)(2,9)(4,21)(5,27)(6,19)(7,17)(8,18)(10,35)(11,38)(12,41)(13,34)(14,31)(15,25)(16,28)(20,48)(22,42)(23,44)(24,43)(26,45)(29,46)(30,47)(32,37)(33,39)(36,61)(40,56)(49,58)(50,59)(51,53)(52,64)(54,57)(55,63)(60,62);(1,4,7,25,8,24,6,23)(2,10,12,37,13,20,31,50)(3,15,19,21,18,44,17,43)(5,22,30,54,29,55,28,53)(9,32,14,35,34,59,41,48)(11,36,39,62,40,49,26,52)(16,42,46,51,47,63,27,57)(33,58,38,60,45,61,56,64);(1,5,6,28,8,29,7,30)(2,11,12,39,13,40,31,26)(3,16,17,27,18,47,19,46)(4,22,23,53,24,55,25,54)(9,33,14,38,34,45,41,56)(10,36,37,62,20,49,50,52)(15,42,43,57,44,63,21,51)(32,58,35,60,59,61,48,64)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264928, #72264989
maximal normal subgroup of	#72264928, #72264989
maximal quotient of	#72265738

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293234);