#374021

order	512
nilpotency class	3
isoclinism	isoclinic class of #374018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14134
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2378 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341332, #72341353, #72341355
maximal normal subgroup of	#72341332, #72341353, #72341355

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281217);