#404674

order	512
nilpotency class	3
isoclinism	isoclinic class of #396612
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2850
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14898
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

892 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)(3,9,16,31)(4,10,21,35)(5,24)(6,13)(7,27)(8,12)(11,25)(14,29)(15,42,43,56)(17,32,18,45)(19,33)(20,48,49,60)(22,36,23,51)(26,38)(28,37)(30,46,55,44)(34,52,59,50)(39,61)(40,54)(41,58)(47,53)(57,62)(63,64);(1,3)(2,9,12,31)(4,19,21,41)(5,15)(6,17)(7,18)(8,16)(10,33)(11,30,37,55)(13,32,27,45)(14,22,40,23)(20,47,49,63)(24,42,38,56)(25,44)(26,43)(28,46)(29,51)(34,57)(35,58)(36,54)(39,50,62,52)(48,53)(59,61)(60,64);(1,4)(2,10,12,35)(3,14,16,40)(5,20)(6,22)(7,23)(8,21)(9,29)(11,34,37,59)(13,36,27,51)(15,39,43,62)(17,19,18,41)(24,48,38,60)(25,50)(26,49)(28,52)(30,53)(31,54)(32,33)(42,61)(44,47,46,63)(45,58)(55,64)(56,57);(1,5,6,25,8,26,7,28)(2,11,13,38,12,37,27,24)(3,15,17,44,16,43,18,46)(4,20,22,50,21,49,23,52)(9,30,32,56,31,55,45,42)(10,34,36,60,35,59,51,48)(14,39,19,47,40,62,41,63)(29,53,33,57,54,64,58,61)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311870);