#147203

order	512
nilpotency class	4
isoclinism	isoclinic class of #13175
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1943
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10881
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

495 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)(3,17,21,48)(4,23)(5,14)(6,15)(7,33)(8,13)(9,16)(10,19,37,55)(11,26)(12,29)(18,49,52,64)(20,56,31,38)(22,44,54,36)(24,59,51,47)(25,61)(27,46)(28,45)(30,43)(32,42)(34,41)(35,57,58,53)(39,60,63,50)(40,62);(1,3,7,22,6,31,30,55,9,21,29,54,8,20,5,19)(2,10,14,38,13,44,12,17,16,37,43,56,15,36,33,48)(4,24,27,60,32,18,25,53,28,51,34,50,26,52,62,57)(11,39,41,47,45,35,61,49,42,63,46,59,23,58,40,64);(1,4)(2,11)(3,18,21,52)(5,25)(6,32)(7,34)(8,26)(9,28)(10,35,37,58)(12,40)(13,45)(14,46)(15,23)(16,42)(17,47,48,59)(19,50,55,60)(20,24,31,51)(22,57,54,53)(27,29)(30,62)(33,61)(36,39,44,63)(38,64,56,49)(41,43)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^b=fi,e^c=ei,g^c=gi

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, 54399);