#499443

order	512
nilpotency class	4
isoclinism	isoclinic class of #13522
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2088
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#2088
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

551 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,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,g^b=gi,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320565, #72320632, #72320665
maximal normal subgroup of	#72320565, #72320632, #72320665
maximal quotient of	#72322826

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406639);