#149117

order	512
nilpotency class	4
isoclinism	isoclinic class of #13497
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2075
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

417 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)(3,16,21,48)(4,23)(5,13)(6,33)(7,35)(8,36)(9,15)(10,18,40,20)(11,26)(12,32)(14,31)(17,38,51,61)(19,47,30,41)(22,54,37,39)(24,57,49,63)(25,44)(27,56)(28,60)(29,46)(34,45)(42,53,64,50)(43,59)(52,55,58,62);(1,3)(2,10,15,40)(4,24,29,49)(5,30)(6,20)(7,19)(8,37)(9,21)(11,42)(12,39,33,54)(13,16,35,48)(14,47,36,41)(17,27,51,25)(18,32)(22,31)(23,55)(26,58,34,52)(28,50,59,53)(38,60)(43,61)(44,57)(45,64)(46,62)(56,63);(1,4,5,25,8,28,32,26,9,29,7,27,31,59,6,34)(2,11,12,43,14,44,13,46,15,45,33,60,36,56,35,23)(3,17,18,49,22,52,30,50,21,51,20,24,37,58,19,53)(10,38,16,42,41,62,39,63,40,61,48,64,47,55,54,57)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=dh,e^b=ehi,f^b=fh,g^b=gi,h^b=hi,e^c=ei

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, 56313);