#148541

order	512
nilpotency class	4
isoclinism	isoclinic class of #13389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

630 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)(5,6,11,17,12,19,37,18)(7,20,24,50,25,52,38,40)(8,26,31,46,32,53,65,45)(9,16,33,15,34,51,10,35)(13,23,41,22,43,59,42,60)(14,30,47,36,49,28,48,29)(21,39,57,62,58,67,64,66)(27,63,61,44,55,68,56,54);(5,7,12,25)(6,13,19,43)(8,27,32,55)(9,23,34,59)(10,22,33,60)(11,38,37,24)(14,44,49,54)(15,50,35,40)(16,52,51,20)(17,42,18,41)(21,28,58,30)(26,62,53,66)(29,57,36,64)(31,56,65,61)(39,45,67,46)(47,63,48,68);(2,4)(5,8,9,28,11,31,33,29,12,32,34,30,37,65,10,36)(6,14,15,45,18,48,16,53,19,49,35,46,17,47,51,26)(7,21,22,55,24,57,59,56,25,58,60,27,38,64,23,61)(13,39,20,54,42,66,40,68,43,67,52,44,41,62,50,63)

Polycyclic presentation:

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

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, 55737);