#147439

order	512
nilpotency class	4
isoclinism	isoclinic class of #13156
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1936
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 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)(5,6,10,17,13,19,12,39)(7,20,24,43,26,50,23,54)(8,27,30,49,32,48,37,15)(9,33,35,51,36,18,11,16)(14,25,42,34,44,61,41,22)(21,55,28,53,60,65,58,68)(29,46,38,52,56,67,31,47)(40,63,64,62,66,57,45,59);(2,4)(5,7,11,25,12,23,36,22,13,26,35,61,10,24,9,34)(6,14,18,43,17,42,16,50,19,44,33,54,39,41,51,20)(8,28,31,63,30,60,29,62,32,58,38,57,37,21,56,59)(15,45,47,53,48,66,67,55,49,64,52,68,27,40,46,65);(5,8,13,32)(6,15,19,49)(7,21)(9,29,36,56)(10,37,12,30)(11,38,35,31)(14,40)(16,46,51,67)(17,48,39,27)(18,52,33,47)(20,53)(22,57)(23,28)(24,58)(25,62)(26,60)(34,63)(41,64)(42,45)(43,55)(44,66)(50,68)(54,65)(59,61)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54635);