#111169

order	512
nilpotency class	3
isoclinism	isoclinic class of #111151
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di

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, 18365);