#236418

order	512
nilpotency class	3
isoclinism	isoclinic class of #236399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1118 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)(3,13)(4,18)(5,10)(6,11)(7,12)(8,16)(9,20)(14,25)(15,24)(17,23)(19,26)(21,27)(22,30)(28,29)(31,32)(33,34)(35,43)(36,40)(37,49)(38,42)(39,46)(41,52)(44,60)(45,58)(47,59)(48,57)(50,54)(51,56)(53,63)(55,62)(61,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,25)(10,13)(11,23)(12,24)(18,28)(19,29)(20,31)(21,30)(22,26)(27,32)(33,35,38,45)(34,39,42,54)(36,44,48,61)(37,50,52,46)(40,55,57,53)(41,58,49,43)(47,63,51,62)(56,64,59,60);(1,4,7,21)(2,9,12,26)(3,14,17,30)(5,20,6,19)(8,22,24,25)(10,27,11,18)(13,28,23,32)(15,31,16,29)(33,36,38,48)(34,40,42,57)(35,44,45,61)(37,51,52,47)(39,53,54,55)(41,59,49,56)(43,60,58,64)(46,63,50,62);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,24,23)(9,18,26,27)(14,29,30,31)(22,32,25,28)(35,45)(39,54)(43,58)(44,61)(46,50)(53,55)(60,64)(62,63)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,d^c=dhi,e^c=ei,f^c=fhi

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, 143614);