#199818

order	512
nilpotency class	3
isoclinism	isoclinic class of #199815
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

960 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)(14,28)(15,24)(17,23)(19,25)(21,26)(22,30)(27,29)(31,32)(33,34)(35,39,45,53)(36,40)(37,41)(38,42)(43,51,60,63)(44,52,48,56)(46,54)(47,55)(49,57)(50,58)(59,62,61,64);(1,3,5,16,7,17,6,15)(2,8,11,13,12,24,10,23)(4,14,20,29,21,30,19,31)(9,22,18,27,25,28,26,32)(33,35)(34,39,42,53)(36,43)(37,48)(38,45)(40,51,55,63)(41,56,58,52)(44,50)(46,61)(47,60)(49,59)(54,64,57,62);(1,4,7,21)(2,9,12,25)(3,14,17,30)(5,20,6,19)(8,22,24,28)(10,26,11,18)(13,27,23,32)(15,31,16,29)(33,36,38,47)(34,40,42,55)(35,43,45,60)(37,49,50,46)(39,51,53,63)(41,57,58,54)(44,61,48,59)(52,64,56,62);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,24,23)(9,18,25,26)(14,29,30,31)(22,32,28,27)(33,37,38,50)(34,41,42,58)(35,44,45,48)(36,46,47,49)(39,52,53,56)(40,54,55,57)(43,59,60,61)(51,62,63,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi,d^c=dh,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107014);