#478940

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17176
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

661 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)(3,14)(4,19)(5,24,27,38)(6,28)(7,12)(8,13)(9,17)(10,21)(11,26,39,30)(15,42)(16,37,47,61)(18,33)(20,35,54,63)(22,36)(23,51)(25,32,48,41)(29,34)(31,55)(40,45)(43,59,46,60)(44,49)(50,53,57,56)(52,64,58,62);(1,3,6,18,8,17,7,29)(2,9,12,33,13,14,28,34)(4,15,21,45,23,44,22,55)(5,25,26,16,27,48,30,47)(10,31,19,49,36,40,51,42)(11,37,38,32,39,61,24,41)(20,52,53,43,54,58,56,46)(35,62,50,59,63,64,57,60);(1,4,6,21,8,23,7,22)(2,10,12,19,13,36,28,51)(3,15,18,45,17,44,29,55)(5,20,26,53,27,54,30,56)(9,31,33,49,14,40,34,42)(11,35,38,50,39,63,24,57)(16,46,48,52,47,43,25,58)(32,60,61,62,41,59,37,64);(1,5)(2,11)(3,16,17,47)(4,20)(6,26)(7,30)(8,27)(9,32,14,41)(10,35)(12,38)(13,39)(15,43,44,46)(18,48,29,25)(19,50)(21,53)(22,56)(23,54)(24,28)(31,59,40,60)(33,61,34,37)(36,63)(42,62,49,64)(45,58,55,52)(51,57)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386136);