#453024

order	512
nilpotency class	3
isoclinism	isoclinic class of #424595
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16076
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

741 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360220);