#453838

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16146
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

636 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)(3,14)(4,20,25,50)(5,11)(6,30)(7,12)(8,13)(9,17)(10,24,35,23)(15,41,21,31)(16,45)(18,33)(19,39)(22,49,55,64)(26,56)(27,58)(28,37)(29,38)(32,46)(34,54,61,53)(36,47)(40,44,48,43)(42,59,51,62)(52,60,57,63);(1,3,7,17,8,19,6,18)(2,9,12,39,13,33,30,14)(4,21,24,44,25,15,23,43)(5,26,28,47,29,46,27,16)(10,31,20,48,35,41,50,40)(11,36,37,32,38,45,58,56)(22,52,54,42,55,57,53,51)(34,60,49,59,61,63,64,62);(1,4,6,23,8,25,7,24)(2,10,12,20,13,35,30,50)(3,15,17,43,19,21,18,44)(5,22,27,53,29,55,28,54)(9,31,14,40,33,41,39,48)(11,34,37,49,38,61,58,64)(16,42,26,57,47,51,46,52)(32,59,36,60,56,62,45,63);(1,5,7,28,8,29,6,27)(2,11,12,37,13,38,30,58)(3,16,18,46,19,47,17,26)(4,22,24,54,25,55,23,53)(9,32,14,36,33,56,39,45)(10,34,20,49,35,61,50,64)(15,42,44,52,21,51,43,57)(31,59,40,60,41,62,48,63)

Polycyclic presentation:

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

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, 361034);