#146454

order	512
nilpotency class	4
isoclinism	isoclinic class of #13068
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1893
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10721
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

861 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=di,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314266
maximal normal subgroup of	#72314266

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53650);