#194463

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

983 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)(3,12,15,24)(4,16)(5,11)(6,20)(7,10)(8,14,23,19)(9,18)(13,28,29,22)(17,25)(21,26)(27,30,32,31)(33,34)(35,45,48,58)(36,49,52,41)(37,42)(38,43)(39,44)(40,47,55,53)(46,59)(50,57,51,56)(54,60)(61,64)(62,63);(1,3,5,19,7,15,6,14)(2,8,11,24,10,23,20,12)(4,13,18,31,17,29,21,30)(9,22,25,32,26,28,16,27)(33,35,37,47,39,48,38,53)(34,40,42,45,44,55,43,58)(36,46,51,62,52,61,50,60)(41,54,57,59,49,63,56,64);(1,4)(2,9)(3,13)(5,18)(6,21)(7,17)(8,22)(10,26)(11,25)(12,27)(14,30)(15,29)(16,20)(19,31)(23,28)(24,32)(33,36)(34,41)(35,46)(37,51)(38,50)(39,52)(40,54)(42,57)(43,56)(44,49)(45,59)(47,62)(48,61)(53,60)(55,63)(58,64);(2,10)(4,17)(8,23)(11,20)(12,24)(13,29)(18,21)(30,31)(33,37,39,38)(34,42,44,43)(35,47,48,53)(36,50,52,51)(40,45,55,58)(41,56,49,57)(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,e^b=eh,f^b=fi,g^b=gh,d^c=dh,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, 101659);