#403906

order	512
nilpotency class	3
isoclinism	isoclinic class of #394813
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2436
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14884
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

837 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=g,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,f^b=fi,f^c=fi,h^c=hi,f^d=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, 311102);