#148954

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2054
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11256
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

431 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=di,f^b=fh,g^b=gi,h^b=hi,f^c=fhi,h^c=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, 56150);