#148928

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1924
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

460 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=di,f^c=fi,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, 56124);