#198138

order	512
nilpotency class	3
isoclinism	isoclinic class of #198135
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

976 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)(4,17,20,25)(5,10)(6,11)(7,12)(8,15)(9,19,26,18)(14,28,30,22)(16,23)(21,24)(27,29,32,31)(33,34)(35,39,45,53)(36,40)(37,48)(38,42)(41,50)(43,51,58,63)(44,59,61,52)(46,56)(47,55)(49,54)(57,64,60,62);(1,3,5,15,7,16,6,21)(2,8,11,13,12,24,10,23)(4,14,19,31,20,30,18,29)(9,22,25,27,26,28,17,32)(33,35)(34,39,42,53)(36,43)(37,44)(38,45)(40,51,55,63)(41,52,48,59)(46,57)(47,58)(49,60)(50,61)(54,62,56,64);(1,4,7,20)(2,9,12,26)(3,14,16,30)(5,19,6,18)(8,22,24,28)(10,17,11,25)(13,27,23,32)(15,31,21,29)(33,36,38,47)(34,40,42,55)(35,43,45,58)(37,49,50,46)(39,51,53,63)(41,56,48,54)(44,60,61,57)(52,64,59,62);(1,5,7,6)(2,10,12,11)(3,15,16,21)(4,18,20,19)(8,23,24,13)(9,25,26,17)(14,29,30,31)(22,27,28,32)(33,37,38,50)(34,41,42,48)(35,44,45,61)(36,46,47,49)(39,52,53,59)(40,54,55,56)(43,57,58,60)(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,d^a=di,e^a=eh,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,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, 105334);