#463920

order	512
nilpotency class	3
isoclinism	isoclinic class of #459490
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2733
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16496
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

660 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=f,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,h^c=hi,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, 371116);