#148875

order	512
nilpotency class	4
isoclinism	isoclinic class of #134691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1807
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

472 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,g^b=gi

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, 56071);