#113633

order	512
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
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

745 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)(3,12,16,8)(4,17)(5,22)(6,11)(7,10)(9,20)(13,30,31,25)(14,24,15,27)(18,23,32,29)(19,28)(21,26)(33,34,38,42,39,43,37,44)(35,45,47,40,48,56,54,57)(36,49,52,60,53,41,51,59)(46,58,50,64,61,63,62,55);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,25)(10,12)(11,27)(13,21)(17,23)(19,32)(20,31)(22,24)(26,30)(28,29)(33,35)(34,40)(36,50)(37,54)(38,47)(39,48)(41,58)(42,56)(43,57)(44,45)(46,51)(49,63)(52,61)(53,62)(55,60)(59,64);(1,4,7,19)(2,9,10,26)(3,13,16,31)(5,21,6,20)(8,23,12,29)(11,28,22,17)(14,18,15,32)(24,30,27,25)(33,36,39,53)(34,41,43,49)(35,46,48,61)(37,52,38,51)(40,55,57,64)(42,60,44,59)(45,58,56,63)(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,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,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, 20829);