#463810

order	512
nilpotency class	3
isoclinism	isoclinic class of #455397
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16490
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

557 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=ef,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,g^b=gi,d^c=de,g^c=gi,h^c=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, 371006);