#194448

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
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

980 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)(3,12,15,23)(4,16)(5,10)(6,20)(7,11)(8,14,24,19)(9,17)(13,28,31,22)(18,25)(21,26)(27,29,32,30)(33,34)(35,44,47,55)(36,48,50,41)(37,51)(38,42)(39,43)(40,46,56,52)(45,59)(49,57,53,58)(54,61)(60,63)(62,64);(1,3,5,19,7,15,6,14)(2,8,10,23,11,24,20,12)(4,13,17,30,18,31,21,29)(9,22,25,32,26,28,16,27)(33,35,37,46,39,47,38,52)(34,40,51,44,43,56,42,55)(36,45,49,60,50,62,53,61)(41,54,58,59,48,63,57,64);(1,4)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,25)(11,26)(12,27)(14,29)(15,31)(16,20)(19,30)(23,32)(24,28)(33,36)(34,41)(35,45)(37,49)(38,53)(39,50)(40,54)(42,57)(43,48)(44,59)(46,60)(47,62)(51,58)(52,61)(55,64)(56,63);(33,37,39,38)(34,42,43,51)(35,46,47,52)(36,49,50,53)(40,55,56,44)(41,57,48,58)(45,60,62,61)(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=di,f^a=fh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,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, 101644);