#187378

order	512
nilpotency class	3
isoclinism	isoclinic class of #187204
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62826	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2028 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72272561, #72272564, #72272576
maximal normal subgroup of	#72272561, #72272564, #72272576

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94574);