#478941

order	512
nilpotency class	3
isoclinism	isoclinic class of #478348
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17176
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

669 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)(3,13)(4,18)(5,22,25,10)(6,26)(7,12)(8,16)(9,20)(11,27)(14,38)(15,36,43,29)(17,30)(19,49,52,47)(21,48)(23,33,44,54)(24,34,58,56)(28,59)(31,51,55,64)(32,53)(35,41)(37,45)(39,62,42,63)(40,46)(50,60,57,61);(1,3,6,17,7,16,27,45)(2,8,11,30,12,13,26,37)(4,14,20,41,21,40,53,59)(5,23,24,15,25,44,58,43)(9,28,18,46,32,35,48,38)(10,33,34,29,22,54,56,36)(19,50,51,39,52,57,64,42)(31,62,47,60,55,63,49,61);(1,4,6,20,7,21,27,53)(2,9,11,18,12,32,26,48)(3,14,17,41,16,40,45,59)(5,19,24,51,25,52,58,64)(8,28,30,46,13,35,37,38)(10,31,34,47,22,55,56,49)(15,42,44,50,43,39,23,57)(29,61,54,62,36,60,33,63);(1,5)(2,10)(3,15,16,43)(4,19)(6,24)(7,25)(8,29,13,36)(9,31)(11,34)(12,22)(14,39,40,42)(17,44,45,23)(18,47)(20,51)(21,52)(26,56)(27,58)(28,60,35,61)(30,54,37,33)(32,55)(38,62,46,63)(41,57,59,50)(48,49)(53,64)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386137);