#194466

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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

975 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)(3,12,14,24)(4,16)(5,10)(6,20)(7,11)(8,15,23,19)(9,18)(13,28,31,22)(17,26)(21,25)(27,29,32,30)(33,34)(35,44,48,57)(36,49,52,41)(37,53)(38,42)(39,43)(40,47,55,46)(45,59)(50,56,51,58)(54,61)(60,63)(62,64);(1,3,5,19,7,14,6,15)(2,8,10,24,11,23,20,12)(4,13,18,30,17,31,21,29)(9,22,26,32,25,28,16,27)(33,35,37,47,39,48,38,46)(34,40,53,44,43,55,42,57)(36,45,51,60,52,62,50,61)(41,54,56,59,49,63,58,64);(1,4)(2,9)(3,13)(5,18)(6,21)(7,17)(8,22)(10,26)(11,25)(12,27)(14,31)(15,29)(16,20)(19,30)(23,28)(24,32)(33,36)(34,41)(35,45)(37,51)(38,50)(39,52)(40,54)(42,58)(43,49)(44,59)(46,61)(47,60)(48,62)(53,56)(55,63)(57,64);(3,14)(4,17)(8,23)(9,25)(12,24)(15,19)(16,26)(18,21)(33,37,39,38)(34,42,43,53)(35,46,48,47)(36,50,52,51)(40,44,55,57)(41,56,49,58)(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,d^b=dh,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101662);