#147924

order	512
nilpotency class	4
isoclinism	isoclinic class of #136735
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

958 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)(7,19,23,48)(8,14)(9,16)(10,32)(11,34)(12,18)(13,21,39,57)(15,30)(17,33)(20,53,55,41)(22,51,58,40)(24,46,35,38)(25,60,61,37)(26,43)(27,49)(28,52)(29,45)(31,50)(36,44)(42,62)(47,63)(54,66,64,68)(56,67,59,65);(5,7)(6,13)(8,25)(9,22)(10,21)(11,35)(12,23)(14,41)(15,46)(16,48)(17,51)(18,39)(19,50)(20,27)(24,33)(26,59)(28,64)(29,61)(30,57)(31,58)(32,38)(34,40)(36,54)(37,43)(42,66)(44,65)(45,53)(47,60)(49,68)(52,67)(55,62)(56,63);(2,4)(5,8,12,29)(6,14,18,45)(7,20,23,55)(9,27,31,62)(10,26,30,63)(11,36,33,28)(13,37,39,60)(15,47,32,43)(16,49,50,42)(17,52,34,44)(19,53,48,41)(21,25,57,61)(22,54,58,64)(24,59,35,56)(38,67,46,65)(40,68,51,66)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dhi,f^c=fh,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, 55120);