#147692

order	512
nilpotency class	4
isoclinism	isoclinic class of #13391
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10969
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1001 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)(3,14,19,44)(4,20,24,41)(5,26)(6,11)(7,13)(8,16,34,51)(9,23,37,25)(10,28)(12,30)(15,45)(17,43,18,38)(21,42)(22,54,57,36)(27,33,50,32)(29,39)(31,59)(35,46)(40,49)(47,64)(48,62)(52,56,63,58)(53,61)(55,60);(1,3)(2,8)(4,21)(5,27)(6,17)(7,19)(9,35)(10,38)(11,32)(12,33)(13,34)(14,39)(15,23)(16,49)(18,30)(20,31)(22,55)(24,47)(25,48)(26,44)(28,51)(29,50)(36,42)(37,60)(40,43)(41,61)(45,52)(46,57)(53,56)(54,64)(58,59)(62,63);(1,4)(2,9)(3,15)(5,22)(6,25)(7,24)(8,31)(10,36)(11,41)(12,20)(13,37)(14,42)(16,46)(17,21)(18,47)(19,48)(23,30)(26,52)(27,53)(28,58)(29,57)(32,35)(33,60)(34,61)(38,45)(39,63)(40,54)(43,62)(44,64)(49,56)(50,59)(51,55)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54888);