#148396

order	512
nilpotency class	4
isoclinism	isoclinic class of #13402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
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:

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

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=fhi,h^a=hi,c^b=cf,e^b=ei,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, 55592);