#479772

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17239
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

603 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)(3,15)(4,20)(5,11,29,41)(6,13)(7,12)(8,14)(9,18)(10,23)(16,45)(17,49,25,44)(19,42)(21,38,55,61)(22,36)(24,35)(26,34,54,62)(27,40,28,39)(30,33)(31,48)(32,52,37,51)(43,47)(46,58,53,60)(50,64,57,59)(56,63);(1,3,7,18,8,19,6,30)(2,9,12,42,14,33,13,15)(4,16,23,47,24,48,22,56)(5,25,28,52,29,17,27,51)(10,31,35,63,36,45,20,43)(11,37,39,49,41,32,40,44)(21,53,54,57,55,46,26,50)(34,60,38,59,62,58,61,64);(1,4,7,23,8,24,6,22)(2,10,13,20,14,36,12,35)(3,16,18,47,19,48,30,56)(5,26,28,21,29,54,27,55)(9,31,15,43,33,45,42,63)(11,38,40,34,41,61,39,62)(17,50,51,53,25,57,52,46)(32,59,49,60,37,64,44,58);(1,5)(2,11)(3,17)(4,21)(6,28)(7,27)(8,29)(9,32)(10,34)(12,40)(13,39)(14,41)(15,44)(16,46)(18,52)(19,25)(20,38)(22,54)(23,26)(24,55)(30,51)(31,58)(33,37)(35,61)(36,62)(42,49)(43,64)(45,60)(47,57)(48,53)(50,56)(59,63)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,e^d=ei,f^d=fi,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, 386968);