#478820

order	512
nilpotency class	3
isoclinism	isoclinic class of #478819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17170
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

628 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386016);