#148419

order	512
nilpotency class	4
isoclinism	isoclinic class of #13523
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2092
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

656 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:

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

Polycyclic presentation:

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

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, 55615);