#148377

order	512
nilpotency class	4
isoclinism	isoclinic class of #135930
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 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)(7,19,23,51)(8,14)(9,16)(10,32)(11,35)(12,18)(13,21,39,56)(15,30)(17,34)(20,41,53,67)(22,40,33,62)(24,38,58,47)(25,37,55,64)(26,43)(27,48)(28,50)(29,45)(31,49)(36,44)(42,60)(46,61)(52,65,57,66)(54,68,59,63);(5,7,9,21,11,24,30,22,12,23,31,56,34,58,10,33)(6,13,15,19,17,40,16,47,18,39,32,51,35,62,49,38)(8,25,26,53,28,57,60,59,29,55,61,20,36,52,27,54)(14,41,42,37,44,63,43,65,45,67,48,64,50,68,46,66);(2,4)(5,8,12,29)(6,14,18,45)(7,20,23,53)(9,27,31,60)(10,26,30,61)(11,36,34,28)(13,37,39,64)(15,46,32,43)(16,48,49,42)(17,50,35,44)(19,41,51,67)(21,55,56,25)(22,57,33,52)(24,59,58,54)(38,63,47,68)(40,66,62,65)

Polycyclic presentation:

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

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, 55573);