#113604

order	512
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20800);