#236693

order	512
nilpotency class	3
isoclinism	isoclinic class of #236683
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

1098 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)(4,17)(5,22)(6,10)(7,11)(8,14)(9,20)(13,30)(15,25)(16,24)(18,29)(19,28)(21,27)(23,31)(26,32)(33,34)(35,40)(36,48)(37,42)(38,52)(39,43)(41,51)(44,59)(45,54)(46,57)(47,55)(49,56)(50,58)(53,62)(60,64)(61,63);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,26)(10,24)(11,25)(12,22)(13,19)(17,23)(20,31)(21,32)(27,30)(28,29)(33,35,39,47)(34,40,43,55)(36,44,50,61)(37,46,38,45)(41,53,56,64)(42,57,52,54)(48,63,58,59)(49,62,51,60);(1,4,7,21)(2,9,11,28)(3,13,16,31)(5,19,6,20)(8,23,25,30)(10,27,22,17)(12,29,24,26)(14,18,15,32)(33,36)(34,41)(35,44)(37,51)(38,49)(39,50)(40,53)(42,58)(43,56)(45,62)(46,60)(47,61)(48,52)(54,63)(55,64)(57,59);(33,37,39,38)(34,42,43,52)(35,45,47,46)(36,49,50,51)(40,54,55,57)(41,48,56,58)(44,60,61,62)(53,59,64,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143889);