#299497

order	512
nilpotency class	3
isoclinism	isoclinic class of #299469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

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, 206693);