#386040

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

1551 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=e,e^2=i,g^2=i,h^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	#72264949, #72264989
maximal normal subgroup of	#72264949, #72264989
maximal quotient of	#72314162

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293236);