#470135

order	512
nilpotency class	3
isoclinism	isoclinic class of #454981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2619
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16676
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

709 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,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, 377331);