#435120

order	512
nilpotency class	3
isoclinism	isoclinic class of #418818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15516
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

749 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 342316);