#194457

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

973 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dhi,f^a=fh,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101653);