#148921

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1850
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:

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

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,c^b=cf,d^b=di,f^b=fi,g^b=gi,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, 56117);