#388378

order	512
nilpotency class	3
isoclinism	isoclinic class of #388376
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14519
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1255 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284899, #72286176, #72286186
maximal normal subgroup of	#72284899, #72286176, #72286186

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295574);