#198918

order	512
nilpotency class	3
isoclinism	isoclinic class of #198843
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

966 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)(3,13,15,23,17,25,16,8)(4,18,19,26,21,9,20,24)(14,27,30,22,29,32,28,31)(33,34)(35,45,47,55)(36,50,53,58)(37,43)(38,44)(39,42)(40,48,56,49)(41,52,57,51)(46,60)(54,61)(59,62)(63,64);(1,3,7,17)(2,8)(4,14,21,29)(5,16,6,15)(9,22)(10,25)(11,13)(12,23)(18,31)(19,30,20,28)(24,32)(26,27)(33,35,37,49,39,47,38,48)(34,40,43,55,42,56,44,45)(36,46,52,62,53,61,51,63)(41,54,50,64,57,60,58,59);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,24)(11,26)(12,18)(13,27)(15,28)(16,30)(17,29)(23,31)(25,32)(33,36)(34,41)(35,46)(37,52)(38,51)(39,53)(40,54)(42,57)(43,50)(44,58)(45,59)(47,61)(48,63)(49,62)(55,64)(56,60);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,21,20)(8,13,23,25)(9,24,18,26)(14,28,29,30)(22,27,31,32)(34,42)(35,47)(41,57)(43,44)(46,61)(48,49)(50,58)(62,63)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=h,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=dhi,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, 106114);