#497049

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22209
prime radical	#575	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

445 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=ce,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224961, C₂×#497049
maximal normal subgroup of	#72224961, C₂×#497049
maximal quotient of	C₂×#497049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404245);