#146679

order	512
nilpotency class	4
isoclinism	isoclinic class of #12705
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10761
prime radical	#3449	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

997 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)(3,13)(4,19)(5,24,25,10)(6,11)(7,12)(8,15,30,16)(9,21,33,20)(14,39)(17,38)(18,29)(22,48)(23,32)(26,52,45,35)(27,36)(28,42)(31,44,56,46)(34,50,58,49)(37,40,53,43)(41,60)(47,55)(51,57)(54,62)(59,61,64,63);(1,3,6,17,7,18,27,47)(2,8,11,31,12,30,36,56)(4,14,22,41,23,42,51,62)(5,15,26,44,25,16,45,46)(9,28,34,54,33,39,58,60)(10,13,35,38,24,29,52,55)(19,37,48,59,32,53,57,64)(20,40,49,61,21,43,50,63);(1,4)(2,9)(3,14)(5,21)(6,22)(7,23)(8,28)(10,19)(11,34)(12,33)(13,37)(15,43)(16,40)(17,41)(18,42)(20,25)(24,32)(26,50)(27,51)(29,53)(30,39)(31,54)(35,48)(36,58)(38,59)(44,63)(45,49)(46,61)(47,62)(52,57)(55,64)(56,60)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317369, #72317378, #72317382, …
maximal normal subgroup of	#72317369, #72317378, #72317382, …
maximal quotient of	C₂×#146679

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53875);