#199809

order	512
nilpotency class	3
isoclinism	isoclinic class of #199806
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

979 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)(8,15)(9,20)(14,28)(16,24)(17,23)(19,25)(21,26)(22,31)(27,30)(29,32)(33,34)(35,39,45,52)(36,40)(37,41)(38,42)(43,50,58,62)(44,51,60,64)(46,53)(47,54)(48,55)(49,56)(57,61,59,63);(1,3,5,15,7,17,6,16)(2,8,11,13,12,24,10,23)(4,14,20,30,21,31,19,29)(9,22,18,27,25,28,26,32)(33,35)(34,39,42,52)(36,43)(37,44)(38,45)(40,50,54,62)(41,51,56,64)(46,57)(47,58)(48,59)(49,60)(53,61,55,63);(1,4,7,21)(2,9,12,25)(3,14,17,31)(5,20,6,19)(8,22,24,28)(10,26,11,18)(13,27,23,32)(15,30,16,29)(33,36,38,47)(34,40,42,54)(35,43,45,58)(37,48,49,46)(39,50,52,62)(41,55,56,53)(44,59,60,57)(51,63,64,61);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,23,24,13)(9,18,25,26)(14,29,31,30)(22,27,28,32)(33,37,38,49)(34,41,42,56)(35,44,45,60)(36,46,47,48)(39,51,52,64)(40,53,54,55)(43,57,58,59)(50,61,62,63)

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,e^b=ei,g^b=gi,d^c=dh,f^c=fhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107005);