#195530

order	512
nilpotency class	3
isoclinism	isoclinic class of #18810
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

981 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)(3,13,17,24)(4,18,21,26)(5,11)(6,12)(7,10)(8,15,23,16)(9,20,25,19)(14,28)(22,31)(27,29)(30,32)(33,34)(35,43,46,57)(36,48,50,40)(37,41)(38,42)(39,47,55,51)(44,53)(45,59)(49,58,52,56)(54,61)(60,64)(62,63);(1,3,5,16,7,17,6,15)(2,8,11,24,10,23,12,13)(4,14,20,29,21,31,19,30)(9,22,18,32,25,28,26,27)(33,35,44,47,38,46,37,51)(34,39,53,43,42,55,41,57)(36,45,49,62,50,60,52,61)(40,54,56,59,48,63,58,64);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,22)(10,25)(11,18)(12,26)(13,27)(15,30)(16,29)(17,31)(23,28)(24,32)(33,36,38,50)(34,40,42,48)(35,45,46,60)(37,52,44,49)(39,54,55,63)(41,58,53,56)(43,59,57,64)(47,62,51,61);(2,10)(8,23)(9,25)(11,12)(13,24)(18,26)(22,28)(27,32)(35,46)(39,55)(43,57)(45,60)(47,51)(54,63)(59,64)(61,62)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dhi,f^a=fi,d^b=di,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh

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, 102726);