#232721

order	512
nilpotency class	3
isoclinism	isoclinic class of #232718
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

1108 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)(4,9)(5,20)(6,11)(7,10)(8,14)(13,29)(15,23)(16,28)(17,22)(18,27)(19,25)(21,26)(24,30)(31,32)(33,34)(35,39)(36,46)(37,49)(38,42)(40,48)(41,51)(43,58)(44,53)(45,62)(47,57)(50,56)(52,60)(54,61)(55,63)(59,64);(1,3,7,16)(2,8,10,23)(4,17,19,31)(5,15,6,14)(9,24,25,29)(11,28,20,12)(13,21,30,18)(22,26,32,27)(33,35)(34,39)(36,43)(37,45)(38,44)(40,55)(41,54)(42,53)(46,64)(47,59)(48,60)(49,62)(50,63)(51,61)(52,56)(57,58);(1,4,7,19)(2,9,10,25)(3,13,16,30)(5,18,6,21)(8,22,23,32)(11,26,20,27)(12,29,28,24)(14,17,15,31)(33,36)(34,40)(35,43)(37,50)(38,47)(39,52)(41,57)(42,56)(44,59)(45,63)(46,49)(48,51)(53,55)(54,64)(58,62)(60,61);(2,10)(8,23)(9,25)(11,20)(12,28)(22,32)(24,29)(26,27)(35,44)(39,53)(43,59)(45,61)(52,55)(54,62)(58,64)(60,63)

Polycyclic presentation:

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

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, 139917);