#146849

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂×C₄²
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

470 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)(5,6,10,17,12,19,35,49,13,20,11,18,9,16,34,48)(7,21,25,51,23,41,56,50,26,39,36,64,24,14,37,40)(8,27,31,59,30,47,53,46,33,44,32,60,29,15,42,45)(22,52,54,66,28,61,62,68,55,38,58,65,57,43,63,67);(5,7,11,25,12,23,34,56,13,26,10,36,9,24,35,37)(6,14,18,40,19,21,48,51,20,41,17,50,16,39,49,64)(8,28,32,62,30,55,42,58,33,57,31,63,29,22,53,54)(15,43,46,67,27,52,60,66,47,61,45,68,44,38,59,65);(5,8,9,29,13,33,12,30)(6,15,16,44,20,47,19,27)(7,22,23,28,26,55,24,57)(10,32,34,53,11,31,35,42)(14,38,21,43,41,52,39,61)(17,46,48,59,18,45,49,60)(25,58,56,63,36,54,37,62)(40,66,51,68,50,65,64,67)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ch,d^b=di,f^b=fi,e^c=ei,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, 54045);