#146957

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1843
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 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)(5,6,9,14,10,15,25,36,11,16,26,37,27,38,51,62)(7,12,19,30,20,31,44,56,18,29,42,54,43,55,64,67)(8,13,23,34,24,35,50,61,22,33,48,59,49,60,66,68)(17,39,41,63,21,45,47,65,40,28,52,53,46,32,57,58);(2,4)(5,7)(6,12)(8,21,22,46)(9,19)(10,20)(11,18)(13,32,33,45)(14,30)(15,31)(16,29)(17,24,40,49)(23,47,48,57)(25,44)(26,42)(27,43)(28,35,39,60)(34,58,59,65)(36,56)(37,54)(38,55)(41,50,52,66)(51,64)(53,61,63,68)(62,67);(5,8)(6,13)(7,17,18,40)(9,23)(10,24)(11,22)(12,28,29,39)(14,34)(15,35)(16,33)(19,41,42,52)(20,21,43,46)(25,50)(26,48)(27,49)(30,53,54,63)(31,32,55,45)(36,61)(37,59)(38,60)(44,47,64,57)(51,66)(56,58,67,65)(62,68)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,f^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ch,e^b=ei,d^c=di,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54153);