#148084

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

648 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)(7,18,23,38)(8,24)(9,14)(10,32)(11,17)(12,20,39,22)(13,27)(15,30)(16,31)(19,49,52,56)(21,37,33,45)(25,59,50,66)(26,57)(28,58)(29,46)(34,41)(35,44)(36,60,67,54)(40,64,65,51)(42,63)(43,62)(47,61)(48,55,68,53);(1,2,3,4)(5,7,11,23)(6,12)(8,25,28,50)(9,20,31,22)(10,33,35,21)(13,40)(14,18)(15,45)(16,38)(17,39)(19,29,52,26)(24,56)(27,54,62,60)(30,55,61,53)(32,48)(34,64,63,51)(36,41)(37,47)(42,67)(43,65)(44,68)(46,66)(49,58)(57,59);(5,8,11,28)(6,13,17,43)(7,19,23,52)(9,29,31,26)(10,34,35,63)(12,36,39,67)(14,42,16,41)(15,46,47,57)(18,40,38,65)(20,25,22,50)(21,54,33,60)(24,44,58,32)(27,61,62,30)(37,49,45,56)(48,59,68,66)(51,55,64,53)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=i,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi,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, 55280);