#441394

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15679
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

801 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,e^c=ei,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, 348590);