#112516

order	512
nilpotency class	3
isoclinism	isoclinic class of #112507
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

741 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)(3,13,17,8)(4,18)(5,12)(6,11)(7,10)(9,20)(14,26,31,32)(15,24,16,25)(19,23,30,29)(21,27)(22,28)(33,34,38,44,39,43,37,42)(35,45,48,40,49,56,47,55)(36,50,51,58,52,41,53,57)(46,60,61,54,62,59,63,64);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,26)(10,13)(11,25)(12,24)(14,21)(18,29)(20,31)(22,30)(23,28)(27,32)(33,35,39,49)(34,40,43,55)(36,46,52,62)(37,48,38,47)(41,54,50,64)(42,56,44,45)(51,63,53,61)(57,60,58,59);(1,4,7,22)(2,9,10,27)(3,14,17,31)(5,21,6,20)(8,23,13,29)(11,28,12,18)(15,19,16,30)(24,26,25,32)(33,36)(34,41)(35,46)(37,51)(38,53)(39,52)(40,54)(42,57)(43,50)(44,58)(45,59)(47,61)(48,63)(49,62)(55,64)(56,60)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19712);