#149042

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2101
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

450 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)(5,6,12,19,36,52,37,53,13,20,9,16,10,17,11,18)(7,21,26,44,24,54,25,14,28,43,23,41,27,42,38,45)(8,29,31,49,32,51,33,55,35,50,39,65,34,15,46,48)(22,47,57,40,60,56,64,67,59,68,61,66,58,62,30,63);(1,2,4,3)(5,7,10,24,13,28,36,27)(6,14,17,41,20,45,52,44)(8,30,32,61,35,64,34,57)(9,25,37,23,12,38,11,26)(15,47,29,62,51,68,50,56)(16,42,53,21,19,54,18,43)(22,31,58,33,59,39,60,46)(40,48,63,49,66,55,67,65);(1,3,4,2)(5,8,9,31,36,32,11,33,13,35,12,39,10,34,37,46)(6,15,16,48,52,29,18,49,20,51,19,55,17,50,53,65)(7,22,23,57,24,60,38,64,28,59,26,61,27,58,25,30)(14,40,21,56,41,67,54,68,45,66,43,62,44,63,42,47)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=i,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dh,f^b=fh,g^b=gh,h^b=hi,e^c=ei,g^c=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, 56238);