#146489

order	512
nilpotency class	4
isoclinism	isoclinic class of #12715
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10725
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#585 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1605 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)(3,14,18,39)(4,20,23,35)(5,24,26,11)(6,12)(7,29)(8,13)(9,16)(10,22,34,28)(15,40)(17,38,19,48)(21,51)(25,55,27,36)(30,58,59,64)(31,44)(32,46)(33,52)(37,47)(41,49,63,60)(42,62)(43,61)(45,57)(50,53)(54,56);(1,3,7,19,8,18,6,17)(2,9,12,31,13,32,29,57)(4,15,22,42,23,43,28,47)(5,16,27,45,26,46,25,44)(10,30,35,60,34,59,20,49)(11,14,36,38,24,39,55,48)(21,41,52,58,53,63,56,64)(33,37,51,61,54,62,50,40);(1,4)(2,10)(3,15)(5,21)(6,28)(7,22)(8,23)(9,30)(11,33)(12,35)(13,34)(14,37)(16,41)(17,47)(18,43)(19,42)(20,29)(24,54)(25,56)(26,53)(27,52)(31,60)(32,59)(36,51)(38,61)(39,62)(40,48)(44,64)(45,58)(46,63)(49,57)(50,55)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317352, #72317399, #72317412
maximal normal subgroup of	#72317352, #72317399, #72317412

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53685);