#471024

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16736
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

788 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:

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

Polycyclic presentation:

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

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, 378220);