#458030

order	512
nilpotency class	3
isoclinism	isoclinic class of #455699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2797
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16308
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

893 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)(6,13)(7,12)(8,14)(9,18)(10,24)(11,28)(16,43)(17,47)(19,33)(20,34)(22,31)(23,54)(25,38)(26,37)(29,39)(30,40)(32,51)(35,45)(36,57)(41,46)(42,50)(44,59)(48,60)(49,58)(52,55)(53,61)(56,62)(63,64);(1,3,6,19,8,18,7,20)(2,9,12,33,14,15,13,34)(4,22,24,16,26,46,25,45)(5,17,28,50,30,49,29,51)(10,35,21,41,38,43,37,31)(11,32,27,58,39,42,40,47)(23,53,55,44,57,64,56,48)(36,61,52,60,54,63,62,59);(1,4,6,24,8,26,7,25)(2,10,12,21,14,38,13,37)(3,16,20,22,18,45,19,46)(5,23,28,55,30,57,29,56)(9,31,34,43,15,41,33,35)(11,36,27,52,39,54,40,62)(17,48,51,64,49,44,50,53)(32,60,47,61,42,59,58,63);(1,5)(2,11)(3,17)(4,23)(6,28)(7,29)(8,30)(9,32)(10,36)(12,27)(13,40)(14,39)(15,42)(16,44)(18,49)(19,50)(20,51)(21,52)(22,53)(24,55)(25,56)(26,57)(31,59)(33,58)(34,47)(35,61)(37,62)(38,54)(41,60)(43,63)(45,48)(46,64)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365226);