#148425

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1862
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,10,16,12,17,11,34)(7,18,22,50,24,51,23,58)(8,14,28,43,29,45,35,44)(9,30,32,47,33,63,19,15)(13,36,38,57,39,52,49,21)(20,48,54,40,55,68,59,65)(25,60,56,37,53,66,64,67)(26,62,61,41,27,46,31,42);(5,7,19,52,11,23,33,57,12,24,32,36,10,22,9,21)(6,13,30,50,16,38,47,51,17,39,63,58,34,49,15,18)(8,25,61,55,28,56,27,59,29,53,31,20,35,64,26,54)(14,40,62,67,44,48,42,66,45,65,46,37,43,68,41,60);(2,4)(5,8)(6,14)(7,20)(9,31)(10,28)(11,35)(12,29)(13,37)(15,46)(16,43)(17,45)(18,48)(19,27)(21,56)(22,54)(23,59)(24,55)(25,52)(26,32)(30,42)(33,61)(34,44)(36,53)(38,66)(39,67)(40,50)(41,63)(47,62)(49,60)(51,68)(57,64)(58,65)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^c=dhi,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55621);