#424969

order	512
nilpotency class	3
isoclinism	isoclinic class of #420424
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2699
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15326
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

940 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)(3,15)(4,21)(5,27,31,42)(6,13)(7,12)(8,14)(9,18)(10,25)(11,30,41,29)(16,32)(17,33,49,57)(19,35)(20,34)(22,36)(23,53,56,37)(24,39)(26,38)(28,44,48,40)(43,46)(45,60,64,63)(47,50)(51,55,62,54)(52,61,58,59);(1,3,6,18,8,20,7,19)(2,9,12,15,14,35,13,34)(4,22,24,47,26,46,25,16)(5,28,29,49,31,48,30,17)(10,36,38,32,39,43,21,50)(11,40,27,57,41,44,42,33)(23,45,54,58,56,64,55,52)(37,61,51,63,53,59,62,60);(1,4,7,25,8,26,6,24)(2,10,13,21,14,39,12,38)(3,16,19,46,20,47,18,22)(5,23,29,54,31,56,30,55)(9,32,34,36,35,50,15,43)(11,37,27,51,41,53,42,62)(17,45,28,58,49,64,48,52)(33,59,40,60,57,61,44,63);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,37)(12,42)(13,27)(14,41)(15,44)(16,45)(18,48)(19,28)(20,49)(21,51)(22,52)(24,55)(25,54)(26,56)(32,59)(34,40)(35,57)(36,60)(38,62)(39,53)(43,63)(46,58)(47,64)(50,61)

Polycyclic presentation:

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

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, 332165);