#148384

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=dhi,e^c=ei,f^c=fh,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, 55580);