#236410

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

1132 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)(33,34,39,43)(35,44,48,59)(36,41,50,58)(37,42,38,51)(40,46,54,47)(45,60,62,64)(49,56,52,57)(53,61,55,63);(1,3,7,17)(2,8,12,24)(4,14,21,30)(5,15,6,16)(9,25,26,22)(10,13,11,23)(18,28,27,32)(19,29,20,31)(33,35,39,48)(34,40,43,54)(36,45,50,62)(37,47,38,46)(41,55,58,53)(42,59,51,44)(49,63,52,61)(56,60,57,64);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,27)(11,18)(12,26)(13,28)(15,31)(16,29)(17,30)(23,32)(24,25)(33,36)(34,41)(35,45)(37,49)(38,52)(39,50)(40,53)(42,56)(43,58)(44,60)(46,61)(47,63)(48,62)(51,57)(54,55)(59,64);(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)(33,37,39,38)(34,42,43,51)(35,46,48,47)(36,49,50,52)(40,44,54,59)(41,56,58,57)(45,61,62,63)(53,60,55,64)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=h,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, 143606);