#147413

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1934
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 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)(7,19)(8,25,28,43)(9,31)(10,16)(11,35)(12,18)(13,21)(14,27,41,30)(15,33)(17,32)(20,48,51,67)(22,45)(23,39)(24,47)(26,59,29,40)(34,42)(36,63,64,68)(37,55)(38,56)(44,50,66,53)(46,54)(49,57,52,65)(58,60,62,61);(2,4)(5,7,11,23,32,54,33,55,12,24,9,21,10,22,34,56)(6,13,15,19,16,37,35,45,18,39,42,47,17,38,31,46)(8,20,29,52,27,50,61,68,28,51,26,49,30,53,60,63)(14,36,40,44,43,65,62,67,41,64,59,66,25,57,58,48);(5,8,10,30,12,28,32,27)(6,14,16,43,18,41,17,25)(7,20,22,53,24,51,54,50)(9,26,33,61,11,29,34,60)(13,36,37,65,39,64,38,57)(15,40,35,62,42,59,31,58)(19,44,45,67,47,66,46,48)(21,49,55,68,23,52,56,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,h^a=hi,e^c=ei,f^c=fi

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, 54609);