#199628

order	512
nilpotency class	3
isoclinism	isoclinic class of #199558
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	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

953 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,f^c=fhi,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, 106824);