#148966

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	#2077
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

436 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)(5,6,9,15,10,18,12,17)(7,19,23,13,24,39,21,37)(8,14,29,43,27,44,26,41)(11,32,30,48,31,16,35,45)(20,50,52,36,54,49,53,65)(22,55,57,66,58,46,33,38)(25,59,56,67,61,68,60,40)(28,63,34,64,62,42,51,47);(5,7,9,21,10,24,12,23)(6,13,17,39,18,37,15,19)(8,25,29,60,27,61,26,56)(11,33,30,58,31,57,35,22)(14,40,41,59,44,67,43,68)(16,46,48,38,32,55,45,66)(20,51,53,28,54,34,52,62)(36,47,49,42,65,64,50,63);(5,8,9,26,10,27,12,29)(6,14,15,41,18,44,17,43)(7,20,21,52,24,54,23,53)(11,34,30,28,31,51,35,62)(13,36,19,49,37,65,39,50)(16,47,45,42,32,64,48,63)(22,56,33,61,58,60,57,25)(38,59,46,67,66,68,55,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fh,g^b=gi,h^b=hi,f^c=fhi,g^c=gi,h^c=hi

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, 56162);