#499394

order	512
nilpotency class	4
isoclinism	isoclinic class of #13504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2077
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2077
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

673 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=e,d^2=i,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320643, #72320647, #72320806
maximal normal subgroup of	#72320643, #72320647, #72320806

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406590);