#453919

order	512
nilpotency class	3
isoclinism	isoclinic class of #427841
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2778
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16223
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

632 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)(3,9,17,33)(4,20)(5,25)(6,14)(7,13)(8,12)(10,22)(11,27)(15,41,43,57)(16,46,26,55)(18,34,19,32)(21,50)(23,37)(24,38)(28,40)(29,39)(30,45,35,44)(31,48,58,47)(36,53)(42,59,51,56)(49,52)(54,60)(61,63,64,62);(1,3,7,18,8,17,6,19)(2,9,14,32,12,33,13,34)(4,15,23,44,24,43,22,45)(5,26,27,47,28,16,29,48)(10,35,38,57,37,30,20,41)(11,31,25,55,39,58,40,46)(21,51,52,62,53,42,54,63)(36,59,49,61,50,56,60,64);(1,4,6,22,8,24,7,23)(2,10,13,20,12,37,14,38)(3,15,19,45,17,43,18,44)(5,21,27,52,28,53,29,54)(9,30,34,57,33,35,32,41)(11,36,25,49,39,50,40,60)(16,42,48,63,26,51,47,62)(31,56,55,64,58,59,46,61);(1,5,6,27,8,28,7,29)(2,11,14,40,12,39,13,25)(3,16,19,48,17,26,18,47)(4,21,22,52,24,53,23,54)(9,31,32,46,33,58,34,55)(10,36,38,60,37,50,20,49)(15,42,45,63,43,51,44,62)(30,56,41,61,35,59,57,64)

Polycyclic presentation:

2^9|a^2=eg,b^2=egh,c^2=efg,d^2=ef,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^b=de,e^b=ei,f^b=fi,h^b=hi,h^d=hi

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, 361115);