#422608

order	512
nilpotency class	3
isoclinism	isoclinic class of #418530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15308
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

962 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)(3,15,19,34)(4,10)(5,25)(6,12)(7,13)(8,14)(9,18,33,29)(11,27)(16,35,45,43)(17,46,47,32)(20,31,50,58)(21,51)(22,37)(23,38)(24,39)(26,56,48,40)(28,41)(30,42)(36,53)(44,57,52,64)(49,62,63,59)(54,60)(55,61);(1,3,6,29,8,19,7,18)(2,9,12,34,14,33,13,15)(4,20,22,16,24,50,23,45)(5,26,27,17,28,48,30,47)(10,35,37,58,39,43,38,31)(11,40,41,32,42,56,25,46)(21,52,53,49,54,44,55,63)(36,57,60,59,61,64,51,62);(1,4,6,22,8,24,7,23)(2,10,12,37,14,39,13,38)(3,16,18,20,19,45,29,50)(5,21,27,53,28,54,30,55)(9,31,15,43,33,58,34,35)(11,36,41,60,42,61,25,51)(17,44,26,49,47,52,48,63)(32,57,40,62,46,64,56,59);(1,5,6,27,8,28,7,30)(2,11,12,41,14,42,13,25)(3,17,18,26,19,47,29,48)(4,21,22,53,24,54,23,55)(9,32,15,40,33,46,34,56)(10,36,37,60,39,61,38,51)(16,44,20,49,45,52,50,63)(31,57,43,62,58,64,35,59)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=e,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 329804);