#198914

order	512
nilpotency class	3
isoclinism	isoclinic class of #198834
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

967 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

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, 106110);