#422712

order	512
nilpotency class	3
isoclinism	isoclinic class of #422711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15309
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1019 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329908);