#459148

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	#16333
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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,e^c=ei,f^c=fi,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, 366344);